B.Sc. Part 1 Mathematics
THREE DIMENSIONAL
CO-ORDINATE GEOMETRY
Ques1: The point A(1, 2, 3) is a vertex of the rectangular parallelopiped formed by the co-ordinate planes and the planes passing through A parallel to the co-ordinate planes. Find the co-ordinates of the other vertices of the parallelopiped. [Kanpur B.Sc. 2000]
Ans: Other vertices of the rectangular parallelopiped are:
O(0,0,0), P(1,0,0), Q(0,2,0), R(0,0,3), B(0,2,3), C(1,0,3) and D(1,2,0).
Ques2: Find the distance between points (2, 0, -5) and (0, 7, -3). [Kanpur B.Sc. 2001]
Ques3: Find the cylindrical co-ordinates of a point whose cartesian co-ordinates are (2,3,4). [Rohil. 2012]
Ques4: Show that the points (0,4,1), (2,3,-1), (4,5,0) and (2,6,2) are the vertices of a square. [Kanpur 1996]
Ques5: Find the co-ordinates of the point which divides the join of (2,3,4) and (3,-4,7) in the ratio 2:-4. [Meerut 2003]
Ans: (1,10,1).
Ques6: Find the direction cosines of the line which is equally inclined to the positive direction of the axes. [Kanpur B.Sc. 1996, 2000, 11; Purvanchal 2013, 14; Bundel. 2013; Gorakh. 2005]
Ques7: Find the direction cosines of the line joining the points (-2,1,-8) and (4,3,-5). [Kashi 2011]
Ans: 6/7, 2/7, 3/7.
Ques8: The projections of a line on the axes are 5, 10, 10. Find its length and direction cosines. [Agra 2008]
Ans: 15; 1/3, 2/3, 2/3.
Ques9: Prove that the points A(2,-1,3), B(4,3,1) and C(3,1,2) are collinear. [Meerut 2010]
Ques10: If A,B,C,D are the points (3,4,5), (4,6,3), (-1,2,4), (1,0,5), find the projection of CD on AB. [Agra 2001]
Ans: 4/3.
Ques11: If P(2,3,-1), Q(3,5,-3), R(1,2,3) and S(3,5,7) are four points, prove by projections that PQ is at right angle to RS. [Purvanchal 2014]
Ques12: If the direction cosines of two lines are related by equations
l+2m+3n = 0 and mn-4nl+3lm = 0, then find the direction cosines of these lines. Also find the angle between these two lines. [Kanpur B.Sc. 2004]
Ques13: Find the angle between the lines whose direction cosines l,m,n satisfy the equations l+m+n=0 and 2lm+2nl-mn=0. [Bundel. 2013; Kanpur B.Sc. 2006, 07; Purvanchal 2008, 10, 12; Avadh 2006; Gorakh. 2005]
Ques14: If the edges of a rectangular parallelopiped are a,b,c, find the angle between any two diagonals of the recatangular parallelopiped. [Kanpur B.Sc. 2002]
Ques15: Prove that if two pairs of opposite edges of a tetrahedron are perpendicular, then the third pair is also perpendicular. [Garhwal 2003]
Ques16: Find the acute angle between the diagonals of a cube. [Agra 2014; Bundel. 1994; Avadh 2011; Garhwal 2001]
Ques17: Show that the lines whose direction cosines are given by the equations 2l+2m-n=0 and mn+nl+lm=0 are at right angles. [Kanpur B.Sc. 2008]
Ques18: Show that the lines whose direction cosines are given by l+m+n=0 and 2mn+3nl-5lm=0, are at right angles. [Bundel. 1993; Meerut 2000, 02, 04, 05, 08; Purvanchal 1995, 2011; Gorakh. 1998; Kumaun 2008]
Ques19: Find the direction cosines of the lines which are given by the equations l+m+n=0 and mn-2nl-2lm=0. [Kanpur B.Sc. 1999; Purvanchal 1998; Gorakh 1997]
Ques20: A plane meets the co-ordinate axes at A,B,C such that the centroid of the triangle ABC is the point (a,b,c). Show that the equation to the plane is x/a+y/b+z/c=3. [Kanpur B.Sc. 2010; Avadh 2010; Kashi 2011; Meerut 2000, 05, 10; Agra 1995]
Ques21: Find the intercepts made on the co-ordinate axes by the plane 5x-3y+2z=8. [Purvanchal 2013; Kanpur B.Sc. 2001, 14]
Ques22: Find the equation of the plane passing through the point (1,-1,2) and parallel to the plane 3x-2y+4z=7. [Kanpur B.Sc. 2005]
Ques23: If P is the point (7,8,9), find the equation to the plane through P at right angle to OP, where O is the origin. [Kanpur B.Sc. 2002]
Ans: 7x+8y+9z=194.
Ques24: Find the equation of the plane passing through the point (1,3,2) which is perpendicular to OP, where O is the origin and the co-ordinates of P are (1,1,1). [Kanpur B.Sc. 2004]
Ans: x+y+z=6.
Ques25: Show that the four points (0,-1,0), (2,1,-1), (1,1,1), (3,3,0) are coplanar. [Kanpur B.Sc. 2000, 11; Bundel. 2013; Agra 1992, 97; Garhwal 1999]
Ques26: Find the co-ordinates of the point, where the line joining (2,-3,1) and (3,-4,-5) cuts the planes 2x+y+z=7. [Bundel. 2012, 14; Rohil. 2008]
Ans: (1,-2,7).
Ques27: Find the intercepts made on the co-ordinate axes by the plane x+2y-2z=9. Find also the direction cosines of the normal to the planes. [Kanpur B.Sc. 2013; Agra 2002; Kumaun 1996]
Ans: 9,9/2,-9/2; 1/3,2/3,-2/3.
Ques28: Find the equations to the planes which pass through the points (0,4,-3), (6,-4,3) and cut off the intercepts form the axes whose sum is zero. [Agra 1994]
Ans: 2x-3y-6z=6, 6x+3y-2z=18.
Ques29: Show that the four points (0,-1,-1), (4,5,1), (3,9,4) and (-4,4,4) lie in a plane. Find also the equation of the plane through them. [Kanpur B.Sc. 2013; Purvanchal 2011]
Ans: 5x-7y+11z+4=0.
Ques30: Find the equation of the plane passing through the points (1,1,0), (1,2,1) and (-2,2,-1). [Kanpur B.Sc. 2013; Agra 2006]
Ans: 2x+3y-3z=5.
Ques31: Find the equation of a plane parallel to the plane x-2y+z+6=0 and passing through the point (1,1,1). [Kashi 2012]
Ans: x-2y+z=0.
Ques32: A variable plane passes through a fixed point (a,b,c) and meets the co-ordinate axes in A,B,C. Show that the locus of the point common to the planes through A,B,C parallel to the co-ordinate planes is
a/x+b/y+c/z=1. [Agra 1995,96; Meerut 2009; Purvanchal 1997; Kumaun 1995]
Ques33: Find the angle between the planes
3x+5y-2z+1=0 and 2x+4y+9z+7=0. [Kanpur B.Sc. 2001]
Ques34: A line which makes angles 45°, 60°, 60° with the positive direction of x-axis, y-axis, z-axis respectively, then find the direction cosines of the line. [Kanpur B.Sc. 2002]
Ques35: Show that the planes 2x-5y+z+2=0 and x+y+3z-1=0 are perpendicular to each other. [Kanpur B.Sc. 2004]
Ques36: Find the equation of the plane passing through the point (1,2,3) and parallel to the plane x-y+z=1. [Kanpur B.Sc. 2003]
Ans: x-y+z=2.
Ques37: Find the equation of the plane passing through the point (-1,3,2) and perpendicular to the planes x+2y+2z=5 and 3x+3y+2z=8. [Purvanchal 2008, 10; Bundel. 1993, 99; Agra 1997, 2014; Garhwal 1992]
Ans: 2x-4y+3z+8=0.
Ques38: Find the equation of the plane passing through (2,3,-4) and (1,-1,3) and parallel to the x-axis. [Kanpur B.Sc. 2007]
Ans: 7y+4z=5.
Ques39: Find the equation of the plane passing through the points (2,2,1) and (9,3,6) and perpendicular to the plane 2x+6y+6z=9. [Kanpur B.Sc. 1997, 99; Bundel. 1995, 98, 2010; Agra 1992; Purvanchal 1993, 95, 2011]
Ans: 3x+4y-5z=9.
Ques40: Show that the plane 2x-3y+z+1=0 is perpendicular to the plane 7x+6y+4z+2=0. [Kanpur B.Sc. 2003]
Ques41: Find the angle between the planes 2x-y+z=7 and x+y+2z=9. [Bundel. 2004, 07]
Ques42: Find the angle between the planes x+y+z+1=0 and 2x-y+4z+4=0. [Rohil. 2012]
Ques43: Find the equation of the plane passing through the point (3,-3,1)and normal to the join the points (3,4,-1) and (2,-1,5). [Kanpur B.Sc. 1990]
Ans: x+5y-6z+18=0.
Ques44: Find the equation of the plane passing through the points (2,-3,1) and (-1,1,-7) and perpendicular to the plane x+2y-5z=1. [Agra 2008; Purvanchal 1999]
Ans: 4x+23y+10z+51=0.
Ques45: Find the equation of the plane through (-1,1,1) and (1,-1,1), and perpendicular to the plane x+2y+2z=5. [Kanpur B.Sc. 2009]
Ans: 2x+2y-3z+3=0.
Ques46: Find the equation of the plane passing through the points (2,-2,1) and (9,3,-6) and perpendicular to the plane 2x+6y+16z+19=0. [Kanpur B.Sc. 2006]
Ans: 61x-63y+16z-264=0.
Ques47: Find the equation of the plane through (1,-1,2) and (2,-2,1) and perpendicular to the plane 6x-2y+2z=9. [Meerut 2005]
Ans: x+y-2z+4=0.
Ques48: Find the equation of the plane passing through the points (1,-2,2), (-3,1,-2) and perpendicular to the plane x+2y-3z=5. [Rohil. 2013]
Ans: x+16y+11z+9=0.
Ques49: Find the equation of the plane passing through the point (2,5,-8) and perpendicular to the planes 2x-3y+4z+1=0 and 4x+y-2z+10=0. [Kanpur B.Sc. 2008]
Ans: x+10y+7z+4=0.
Ques50: Find the equation of the plane passing through the point (2,3,4) and parallel to the plane 5x-6y+7z=3. [Purvanchal 1998; Gorakh. 1997; Kumaun 1994]
Ans: 5x-6y+7z=20.
Ques51: Prove that the points (1,2,3), (0,5,1) are on the same side of the plane y+z-4=0. [Kanpur B.Sc. 2001]
Ques52: Find the distance of the point (1,1,-2) from the plane 3x+4y-12z+8=0. [Kanpur B.Sc. 2004]
Ans: 3.
Ques53: Find the incentre of the tetrahedron formed by the planes x=0, y=0, z=0, x+y+z=a. [Kanpur B.Sc. 1996]
Ques54: Show that the origin lies in the acute angle between the planes x+2y+2z=9 and 4x-3y+12z+13=0. Find the planes bisecting the angles between them and point out which bisects the acute angle. [Kanpur B.Sc. 1995]
Ans: x+35y-10z-156=0.
Ques55: Find the position of two points (1,2,3) and (-1,0,2) with respect to the plane x+y-3z-6=0. [Kashi 2012]
Ans: Same side.
Ques56: Find the shortest distance from the point (2,3,5) on the plane 3x+4y+12z=9. [Rohil. 2010]
Ans: 69/13.
Ques57: Find the distance between the planes
2x-2y+z+3=0 and 4x-4y+2z+5=0. [Bundel. 1993]
Ans: 1/6.
Ques58: Find the distance between the parallel planes 2x-y+3z-4=0 and 6x-3y+9z+13=0. [Avadh 2014; Bundel. 2007]
Ques59: Find the bisector of that angle between the planes 3x-6y+2z+5=0 and 4x-12y+3z-3=0, which contains the origin. [Avadh 2011]
Ans: 67x-162y+47z+44=0.
Ques60: Show that the origin lies in the obtuse angle between the planes 2x-y+2z+3=0 and 3x-2y+6z+8=0. Find the planes bisecting the angles between them and the one which bisects the acute angle. [Purvanchal 2010]
Ans: 5x-y-4z-3=0; 23x-13y+32z+45=0.
Ques61: Find the equation to the plane containing the line of intersection of the planes x+y+z=6 and 2x+3y+4z+5=0, and the point (1,1,1). [Bundel. 2013]
Ans: 20x+23y+26z-69=0.
Ques62: Find the equation to the plane passing through the line of intersection of the planes 2x-7y+4z=3, 3x-5y+4z+11=0, and the point (-2,1,3). [Bundel. 2005,12]
Ans: 15x-47y+28z=7.
Quea63: Find the equation to the plane passing through the line of intersection of the planes 2x-y=0 and 3z-y=0, and perpendicular to the plane 4x+5y-3z=8. [Gorakh. 1999]
Ans: 28x-17y+9z=0.
Ques64: Find the equation to the plane passing through the line of intersection of the planes x+y+z=6, 2x+3y+4z+5=0 and perpendicular to the plane 4x+5y-3z=8. [Rohil. 2009]
Ans: x+7y+13z+96=0.
Ques65: Determine the planes through the intersection of the planes x+y+z=1 and 2x+3y-z+4=0, which are parallel to the three co-ordinate axes. [Rohil.2006; Purvanchal 2009, 12]
Ans: y-3z+6=0, x+4z-7=0, 3x+4y+3=0.
Ques66: Find the area of the triangle included between the plane 2x-3y+4z=12 and the co-ordinate planes. [Kanpur B.Sc. 1999]
Ques67: Prove that the volume of the tetrahedron formed by the four planes: my+nz=0, nz+lx=0, lx+my=0, lx+my+nz=1 is 2/(3lmn). [Kanpur B.Sc. 2007]
Ques68: Prove that the volume of the tetrahedron formed by the planes y+z=0, z+x=0, x+y=0, x+y+z=p. [Kanpur B.Sc. 2008]
Ques69: Find the volume of a tetrahedron in terms of the edges which meet in a point and of the angles which they make with eachother. [Kanpur B.Sc. 1996; Agra 1996]
Ques70: Prove that (a/y-z)+(b/z-x)+(c/x-y)=0 represents a pair of planes. [Kanpur B.Sc. 2006; Bundel. 2012; Rohil. 2010; Gorakh. 2006]
Ques71: Find the area of the triangle whose vertices are (1,2,3), (2,-1,1) and (1,2,-4). [Meerut 2013]
Ques72: Find the area of the triangle included between the plane 3x-4y+z=12 and the co-ordinate planes. [Rohil. 2008; Meerut 2000]
Ques73: A, B, C are the points (3,2,1), (-2,0,-3), (0,0,-2). Find the locus of P if the volume of the tetrahedron PABC is 5. [Agra 1990]
Ans: 2x+3y-4z+22=0.
Ques74: Write equations of straight line in two forms. [Agra 2013]
Ques75: Find the equations of the line passing through the point (2,1,3) and parallel to the line joining the points (-2,3,-4) and (2,5,1). [Kanpur B.Sc. 2003, 12]
Ques76: Find the equations of the line passing through the point (2,1,3) and parallel to the line (x-1)/2 = (y+2)/3 = (z+1)/-1. [Kanpur B.Sc. 2004]
Ques77: Show that the equation of the plane passing through the point (a,b,c) and perpendicular to the line x/l=y/m=z/n is :
(x-a)l + (y-b)m +(z-c)n = 0. [Kanpur B.Sc. 2003]
Ques78: Find the equation of the plane passing through the point (2,3,4) and perpendicular to the line x/-1 = y/2 = z/5. [Kanpur B.Sc. 2005]
Ans: x-2y-5z+24=0.
Ques79: Find the point at which the line (x-1)/2=(y-1)/3=(z-2)/3 meets the plane 2x-3y+z+5=0. [Kanpur B.Sc. 2004, 12, 14]
Ans: (7,10,11).
Ques80: Find the distance of the point (-1,-5,-10) from the point of intersection of the line (x-2)/3=(y+1)/4=(z-2)/12 and the plane x-y+z=5. [Kanpur B.Sc. 1990]
Ans: 13.
Ques81: Find the image of the point (3,5,7) in the plane 2x+y+z=6. [Kanpur 1997, 98]
Ans: (-5,1,3).
Ques82: Find the distance of the point (1,1,1) from the plane 2x-y+2z+3=0. [Kanpur B.Sc. 2003]
Ans: 2.
Ques83: Find the distance of the point (-1,2,5) from the line which passes through (3,4,5) and whose direction cosines are proportional to 2, -3, 6 is 4√61/7. [Kanpur B.Sc. 2000]
Ans: 4/7 √(16).
Ques84: From the point P(1,2,3), PN is drawn perpendicular to the line (x-2)/3=(y-3)/4=(z-4)/5. Find the distance PN, equations of PN and the co-ordinates of N. [Kanpur B.Sc. 2012; Kumaun 1993]
Ques85: Find the equations of the line passing through the points (1,2,3) and (2,3,5). [Kanpur B.Sc. 2005]
Ques86: From the point (1,2,1) perpendicular PN is drawn to the straight line :
(x-3)/3=(y-4)/4=(z-5)/5. Find the co-ordinates of the point N. [Kanpur B.Sc. 2005]
Ans: (24/25, 32/25, 8/5).
Ques87: Find the equation of the perpendicular drawn from the point (3,-1,11) to the line :
x/2=(y-2)/3=(z-3)/4. Also find the co-ordinates of the foot of the perpendicular and the length of the perpendicular. [Kanpur B.Sc. 2006]
Ans: (x-3)/-1=(y+1)/6=(z-11)/-4; (2,5,7); √(53).
Ques88: Find the length of the perpendicular from a given point (1,-2,3) on a straight line through (2,-3,5) which makes equal angles with the axes. [Purvanchal 2009]
Ans: √(14/3).
Ques89: Find the distance of the point (3,7,2) from the line which passes through the point (-1,2,3) and whose direction cosines are proportional to 5,3,1. [Kanpur B.Sc. 2001]
Ans: 3.737 approx.
Ques90: Find the distance of the point (1,1,1) from the line (x-2)/3=(y+3)/2=z/-1. [Bundel. 2013; Kanpur B.Sc. 2010]
Ans: 1/7 √(826).
Ques91: Find the distance of the point (1,1,1) from the line x/1=(y-1)/2=(z-2)/3. [Kanpur B.Sc. 2011]
Ans: 1/7 √(84).
Ques92: Find the equation of the perpendicular from the point (2,4,-1) to the line (x+5)/1=(y+3)/4=(z-6)/-9. Also obtain the foot of the perpendicular. [Purvanchal 2008]
Ans: (x-2)/6=(y-4)/3=(z+1)/2; 7.
Ques93: Find the foot and hence the length of the perpendicular from the point (5,7,3) to the line :
(x-15)/3=(y-29)/8=(5-z)/3. [Purvanchal 2011]
Ques94: Find the distance between the parallel line (x-1)/2=(y-2)/-7=(z-3)/9 and (x-2)/2=(y-3)/-7=(z-4)/9. [Avadh 2008]
Ques95: Find the point where the line (x-3)/1=(y-4)/2=(z-5)/3 meets the plane x+y+z=20. [Rohil. 2013]
Ans: (4/3,2/3,0).
Ans: x+y+z=6.
Ques25: Show that the four points (0,-1,0), (2,1,-1), (1,1,1), (3,3,0) are coplanar. [Kanpur B.Sc. 2000, 11; Bundel. 2013; Agra 1992, 97; Garhwal 1999]
Ques26: Find the co-ordinates of the point, where the line joining (2,-3,1) and (3,-4,-5) cuts the planes 2x+y+z=7. [Bundel. 2012, 14; Rohil. 2008]
Ans: (1,-2,7).
Ques27: Find the intercepts made on the co-ordinate axes by the plane x+2y-2z=9. Find also the direction cosines of the normal to the planes. [Kanpur B.Sc. 2013; Agra 2002; Kumaun 1996]
Ans: 9,9/2,-9/2; 1/3,2/3,-2/3.
Ques28: Find the equations to the planes which pass through the points (0,4,-3), (6,-4,3) and cut off the intercepts form the axes whose sum is zero. [Agra 1994]
Ans: 2x-3y-6z=6, 6x+3y-2z=18.
Ques29: Show that the four points (0,-1,-1), (4,5,1), (3,9,4) and (-4,4,4) lie in a plane. Find also the equation of the plane through them. [Kanpur B.Sc. 2013; Purvanchal 2011]
Ans: 5x-7y+11z+4=0.
Ques30: Find the equation of the plane passing through the points (1,1,0), (1,2,1) and (-2,2,-1). [Kanpur B.Sc. 2013; Agra 2006]
Ans: 2x+3y-3z=5.
Ques31: Find the equation of a plane parallel to the plane x-2y+z+6=0 and passing through the point (1,1,1). [Kashi 2012]
Ans: x-2y+z=0.
Ques32: A variable plane passes through a fixed point (a,b,c) and meets the co-ordinate axes in A,B,C. Show that the locus of the point common to the planes through A,B,C parallel to the co-ordinate planes is
a/x+b/y+c/z=1. [Agra 1995,96; Meerut 2009; Purvanchal 1997; Kumaun 1995]
Ques33: Find the angle between the planes
3x+5y-2z+1=0 and 2x+4y+9z+7=0. [Kanpur B.Sc. 2001]
Ques34: A line which makes angles 45°, 60°, 60° with the positive direction of x-axis, y-axis, z-axis respectively, then find the direction cosines of the line. [Kanpur B.Sc. 2002]
Ques35: Show that the planes 2x-5y+z+2=0 and x+y+3z-1=0 are perpendicular to each other. [Kanpur B.Sc. 2004]
Ques36: Find the equation of the plane passing through the point (1,2,3) and parallel to the plane x-y+z=1. [Kanpur B.Sc. 2003]
Ans: x-y+z=2.
Ques37: Find the equation of the plane passing through the point (-1,3,2) and perpendicular to the planes x+2y+2z=5 and 3x+3y+2z=8. [Purvanchal 2008, 10; Bundel. 1993, 99; Agra 1997, 2014; Garhwal 1992]
Ans: 2x-4y+3z+8=0.
Ques38: Find the equation of the plane passing through (2,3,-4) and (1,-1,3) and parallel to the x-axis. [Kanpur B.Sc. 2007]
Ans: 7y+4z=5.
Ques39: Find the equation of the plane passing through the points (2,2,1) and (9,3,6) and perpendicular to the plane 2x+6y+6z=9. [Kanpur B.Sc. 1997, 99; Bundel. 1995, 98, 2010; Agra 1992; Purvanchal 1993, 95, 2011]
Ans: 3x+4y-5z=9.
Ques40: Show that the plane 2x-3y+z+1=0 is perpendicular to the plane 7x+6y+4z+2=0. [Kanpur B.Sc. 2003]
Ques41: Find the angle between the planes 2x-y+z=7 and x+y+2z=9. [Bundel. 2004, 07]
Ques42: Find the angle between the planes x+y+z+1=0 and 2x-y+4z+4=0. [Rohil. 2012]
Ques43: Find the equation of the plane passing through the point (3,-3,1)and normal to the join the points (3,4,-1) and (2,-1,5). [Kanpur B.Sc. 1990]
Ans: x+5y-6z+18=0.
Ques44: Find the equation of the plane passing through the points (2,-3,1) and (-1,1,-7) and perpendicular to the plane x+2y-5z=1. [Agra 2008; Purvanchal 1999]
Ans: 4x+23y+10z+51=0.
Ques45: Find the equation of the plane through (-1,1,1) and (1,-1,1), and perpendicular to the plane x+2y+2z=5. [Kanpur B.Sc. 2009]
Ans: 2x+2y-3z+3=0.
Ques46: Find the equation of the plane passing through the points (2,-2,1) and (9,3,-6) and perpendicular to the plane 2x+6y+16z+19=0. [Kanpur B.Sc. 2006]
Ans: 61x-63y+16z-264=0.
Ques47: Find the equation of the plane through (1,-1,2) and (2,-2,1) and perpendicular to the plane 6x-2y+2z=9. [Meerut 2005]
Ans: x+y-2z+4=0.
Ques48: Find the equation of the plane passing through the points (1,-2,2), (-3,1,-2) and perpendicular to the plane x+2y-3z=5. [Rohil. 2013]
Ans: x+16y+11z+9=0.
Ques49: Find the equation of the plane passing through the point (2,5,-8) and perpendicular to the planes 2x-3y+4z+1=0 and 4x+y-2z+10=0. [Kanpur B.Sc. 2008]
Ans: x+10y+7z+4=0.
Ques50: Find the equation of the plane passing through the point (2,3,4) and parallel to the plane 5x-6y+7z=3. [Purvanchal 1998; Gorakh. 1997; Kumaun 1994]
Ans: 5x-6y+7z=20.
Ques51: Prove that the points (1,2,3), (0,5,1) are on the same side of the plane y+z-4=0. [Kanpur B.Sc. 2001]
Ques52: Find the distance of the point (1,1,-2) from the plane 3x+4y-12z+8=0. [Kanpur B.Sc. 2004]
Ans: 3.
Ques53: Find the incentre of the tetrahedron formed by the planes x=0, y=0, z=0, x+y+z=a. [Kanpur B.Sc. 1996]
Ques54: Show that the origin lies in the acute angle between the planes x+2y+2z=9 and 4x-3y+12z+13=0. Find the planes bisecting the angles between them and point out which bisects the acute angle. [Kanpur B.Sc. 1995]
Ans: x+35y-10z-156=0.
Ques55: Find the position of two points (1,2,3) and (-1,0,2) with respect to the plane x+y-3z-6=0. [Kashi 2012]
Ans: Same side.
Ques56: Find the shortest distance from the point (2,3,5) on the plane 3x+4y+12z=9. [Rohil. 2010]
Ans: 69/13.
Ques57: Find the distance between the planes
2x-2y+z+3=0 and 4x-4y+2z+5=0. [Bundel. 1993]
Ans: 1/6.
Ques58: Find the distance between the parallel planes 2x-y+3z-4=0 and 6x-3y+9z+13=0. [Avadh 2014; Bundel. 2007]
Ques59: Find the bisector of that angle between the planes 3x-6y+2z+5=0 and 4x-12y+3z-3=0, which contains the origin. [Avadh 2011]
Ans: 67x-162y+47z+44=0.
Ques60: Show that the origin lies in the obtuse angle between the planes 2x-y+2z+3=0 and 3x-2y+6z+8=0. Find the planes bisecting the angles between them and the one which bisects the acute angle. [Purvanchal 2010]
Ans: 5x-y-4z-3=0; 23x-13y+32z+45=0.
Ques61: Find the equation to the plane containing the line of intersection of the planes x+y+z=6 and 2x+3y+4z+5=0, and the point (1,1,1). [Bundel. 2013]
Ans: 20x+23y+26z-69=0.
Ques62: Find the equation to the plane passing through the line of intersection of the planes 2x-7y+4z=3, 3x-5y+4z+11=0, and the point (-2,1,3). [Bundel. 2005,12]
Ans: 15x-47y+28z=7.
Quea63: Find the equation to the plane passing through the line of intersection of the planes 2x-y=0 and 3z-y=0, and perpendicular to the plane 4x+5y-3z=8. [Gorakh. 1999]
Ans: 28x-17y+9z=0.
Ques64: Find the equation to the plane passing through the line of intersection of the planes x+y+z=6, 2x+3y+4z+5=0 and perpendicular to the plane 4x+5y-3z=8. [Rohil. 2009]
Ans: x+7y+13z+96=0.
Ques65: Determine the planes through the intersection of the planes x+y+z=1 and 2x+3y-z+4=0, which are parallel to the three co-ordinate axes. [Rohil.2006; Purvanchal 2009, 12]
Ans: y-3z+6=0, x+4z-7=0, 3x+4y+3=0.
Ques66: Find the area of the triangle included between the plane 2x-3y+4z=12 and the co-ordinate planes. [Kanpur B.Sc. 1999]
Ques67: Prove that the volume of the tetrahedron formed by the four planes: my+nz=0, nz+lx=0, lx+my=0, lx+my+nz=1 is 2/(3lmn). [Kanpur B.Sc. 2007]
Ques68: Prove that the volume of the tetrahedron formed by the planes y+z=0, z+x=0, x+y=0, x+y+z=p. [Kanpur B.Sc. 2008]
Ques69: Find the volume of a tetrahedron in terms of the edges which meet in a point and of the angles which they make with eachother. [Kanpur B.Sc. 1996; Agra 1996]
Ques70: Prove that (a/y-z)+(b/z-x)+(c/x-y)=0 represents a pair of planes. [Kanpur B.Sc. 2006; Bundel. 2012; Rohil. 2010; Gorakh. 2006]
Ques71: Find the area of the triangle whose vertices are (1,2,3), (2,-1,1) and (1,2,-4). [Meerut 2013]
Ques72: Find the area of the triangle included between the plane 3x-4y+z=12 and the co-ordinate planes. [Rohil. 2008; Meerut 2000]
Ques73: A, B, C are the points (3,2,1), (-2,0,-3), (0,0,-2). Find the locus of P if the volume of the tetrahedron PABC is 5. [Agra 1990]
Ans: 2x+3y-4z+22=0.
Ques74: Write equations of straight line in two forms. [Agra 2013]
Ques75: Find the equations of the line passing through the point (2,1,3) and parallel to the line joining the points (-2,3,-4) and (2,5,1). [Kanpur B.Sc. 2003, 12]
Ques76: Find the equations of the line passing through the point (2,1,3) and parallel to the line (x-1)/2 = (y+2)/3 = (z+1)/-1. [Kanpur B.Sc. 2004]
Ques77: Show that the equation of the plane passing through the point (a,b,c) and perpendicular to the line x/l=y/m=z/n is :
(x-a)l + (y-b)m +(z-c)n = 0. [Kanpur B.Sc. 2003]
Ques78: Find the equation of the plane passing through the point (2,3,4) and perpendicular to the line x/-1 = y/2 = z/5. [Kanpur B.Sc. 2005]
Ans: x-2y-5z+24=0.
Ques79: Find the point at which the line (x-1)/2=(y-1)/3=(z-2)/3 meets the plane 2x-3y+z+5=0. [Kanpur B.Sc. 2004, 12, 14]
Ans: (7,10,11).
Ques80: Find the distance of the point (-1,-5,-10) from the point of intersection of the line (x-2)/3=(y+1)/4=(z-2)/12 and the plane x-y+z=5. [Kanpur B.Sc. 1990]
Ans: 13.
Ques81: Find the image of the point (3,5,7) in the plane 2x+y+z=6. [Kanpur 1997, 98]
Ans: (-5,1,3).
Ques82: Find the distance of the point (1,1,1) from the plane 2x-y+2z+3=0. [Kanpur B.Sc. 2003]
Ans: 2.
Ques83: Find the distance of the point (-1,2,5) from the line which passes through (3,4,5) and whose direction cosines are proportional to 2, -3, 6 is 4√61/7. [Kanpur B.Sc. 2000]
Ans: 4/7 √(16).
Ques84: From the point P(1,2,3), PN is drawn perpendicular to the line (x-2)/3=(y-3)/4=(z-4)/5. Find the distance PN, equations of PN and the co-ordinates of N. [Kanpur B.Sc. 2012; Kumaun 1993]
Ques85: Find the equations of the line passing through the points (1,2,3) and (2,3,5). [Kanpur B.Sc. 2005]
Ques86: From the point (1,2,1) perpendicular PN is drawn to the straight line :
(x-3)/3=(y-4)/4=(z-5)/5. Find the co-ordinates of the point N. [Kanpur B.Sc. 2005]
Ans: (24/25, 32/25, 8/5).
Ques87: Find the equation of the perpendicular drawn from the point (3,-1,11) to the line :
x/2=(y-2)/3=(z-3)/4. Also find the co-ordinates of the foot of the perpendicular and the length of the perpendicular. [Kanpur B.Sc. 2006]
Ans: (x-3)/-1=(y+1)/6=(z-11)/-4; (2,5,7); √(53).
Ques88: Find the length of the perpendicular from a given point (1,-2,3) on a straight line through (2,-3,5) which makes equal angles with the axes. [Purvanchal 2009]
Ans: √(14/3).
Ques89: Find the distance of the point (3,7,2) from the line which passes through the point (-1,2,3) and whose direction cosines are proportional to 5,3,1. [Kanpur B.Sc. 2001]
Ans: 3.737 approx.
Ques90: Find the distance of the point (1,1,1) from the line (x-2)/3=(y+3)/2=z/-1. [Bundel. 2013; Kanpur B.Sc. 2010]
Ans: 1/7 √(826).
Ques91: Find the distance of the point (1,1,1) from the line x/1=(y-1)/2=(z-2)/3. [Kanpur B.Sc. 2011]
Ans: 1/7 √(84).
Ques92: Find the equation of the perpendicular from the point (2,4,-1) to the line (x+5)/1=(y+3)/4=(z-6)/-9. Also obtain the foot of the perpendicular. [Purvanchal 2008]
Ans: (x-2)/6=(y-4)/3=(z+1)/2; 7.
Ques93: Find the foot and hence the length of the perpendicular from the point (5,7,3) to the line :
(x-15)/3=(y-29)/8=(5-z)/3. [Purvanchal 2011]
Ques94: Find the distance between the parallel line (x-1)/2=(y-2)/-7=(z-3)/9 and (x-2)/2=(y-3)/-7=(z-4)/9. [Avadh 2008]
Ques95: Find the point where the line (x-3)/1=(y-4)/2=(z-5)/3 meets the plane x+y+z=20. [Rohil. 2013]
Ans: (4/3,2/3,0).
a triangle is formed by lines 2x+3y+z=8,3x+5y+2z=10 and 4x-4y+3z=6 then find the vertices of the triangle
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