Analytic Geometry – Mathematics Optional PYQs (2008–2026)

Here are all 89 Analytic Geometry questions asked in UPSC Mathematics Optional Paper I from 2008 to 2026, arranged under the 3 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

3.1] Coordinates, lines and planes

21 questions

1. Find the points on the lines $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$ and $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ which are nearest to each other. Hence find the shortest distance between the lines and its equation. [2026/15m/2c]

2. Find the shortest distance between the straight lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\mathrm{\text{and}}\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}.$ [2025/10m/2c(ii)]

3. Find the image of the line $x=3-6t,y=2t,z=3+2t$ in the plane $3x+4y-5z+26=0$. [2024/20m/2c]

4. A variable plane which is at a constant distance $3p$ from the origin $O$ cuts the axes in the points $A$, $B$, $C$ respectively. Show that the locus of the centroid of the tetrahedron $OABC$ is $9(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2})=\frac{16}{p^2}.$ [2023/10m/1e]

5. Prove that the locus of a line which meets the lines $y=mx,z=c$; $y=-mx,z=-c$ and the circle $x^2+y^2=a^2,z=0$ is $c^2{m}^2{(cy-mzx)}^2+c^2{(yz-cmx)}^2=a^2{m}^2{(z^2-c^2)}^2.$ [2023/15m/4c]

6. Find the equation of the sphere of smallest possible radius which touches the straight lines : $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$. [2022/15m/3c]

7. Find equation of the plane containing the lines $\begin{array}{cc}\hfill \frac{x+1}{3}\hfill & \hfill =\frac{y+3}{5}=\frac{z+5}{7},\hfill \\ \hfill \frac{x-2}{1}\hfill & \hfill =\frac{y-4}{3}=\frac{z-6}{5}.\hfill \end{array}$ Also find the point of intersection of the given lines. [2021/15m/4c]

8. Show that the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}\mathrm{\text{and}}\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2}$ intersect. Find the coordinates of the point of intersection and the equation of the plane containing them. [2019/10m/1e]

9. Find the projection of the straight line $\frac{x-1}{2}=\frac{y-1}{3}=\frac{z+1}{-1}$ on the plane $x+y+2z=6$. [2018/10m/1e]

10. Find the shortest distance between the lines $\begin{array}{cc}\hfill a_{1}x+b_{1}y+c_{1}z+d_1\hfill & \hfill =0\hfill \\ \hfill a_{2}x+b_{2}y+c_{2}z+d_2\hfill & \hfill =0\hfill \end{array}$ and the $z$-axis. [2018/12m/2d]

11. Find the equation of the plane parallel to $3x-y+3z=8$ and passing through the point $(1,1,1)$. [2018/12m/4d]

12. Find the shortest distance between the skew lines : $\frac{x-3}{3}=\frac{8-y}{1}=\frac{z-3}{1}\mathrm{\text{and}}\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}.$ [2017/10m/1e]

13. Find the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{4}=z-3$ and $y-mx=z=0$. For what value of $m$ will the two lines intersect? [2016/10m/1e]

14. Find the surface generated by a line which intersects the lines $y=a=z$, $x+3z=a=y+z$ and parallel to the plane $x+y=0$. [2016/10m/4a]

15. Obtain the equation of the plane passing through the points $(2,3,1)$ and $(4,-5,3)$ parallel to $x$-axis. [2015/6m/3c(i)]

16. Verify if the lines : $\frac{x-a+d}{\alpha -\delta }=\frac{y-a}{\alpha }=\frac{z-a-d}{\alpha +\delta }\mathrm{\text{and}}\frac{x-b+c}{\beta -\gamma }=\frac{y-b}{\beta }=\frac{z-b-c}{\beta +\gamma }$ are coplanar. If yes, then find the equation of the plane in which they lie. [2015/7m/3c(ii)]

17. Find the equation of the plane which passes through the points $(0,1,1)$ and $(2,0,-1)$, and is parallel to the line joining the points $(-1,1,-2)$, $(3,-2,4)$. Find also the distance between the line and the plane. [2013/10m/1d]

18. Prove that two of the straight lines represented by the equation $x^3+b{x}^{2}y+cx{y}^2+y^3=0$ will be at right angles, if $b+c=-2$. [2012/12m/1e]

19. Find the equations of the straight line through the point $(3,1,2)$ to intersect the straight line $x+4=y+1=2(z-2)$ and parallel to the plane $4x+y+5z=0$. [2011/10m/1e]

20. A line is drawn through a variable point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $z=0$ to meet two fixed lines $y=mx$, $z=c$ and $y=-mx$, $z=-c$. Find the locus of the line. [2009/12m/1e]

21. The plane $x-2y+3z=0$ is rotated through a right angle about its line of intersection with the plane $2x+3y-4z-5=0$; find the equation of the plane in its new position. [2008/12m/1e]

3.2] Sphere, cone and cylinder

46 questions

1. Obtain the equation of the sphere having its centre on the line $5y+2z=0=2x-3y$ and passing through the points $(0,-2,-4)$ and $(2,-1,-1)$. [2026/10m/1e]

2. If $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ represents one of a set of three mutually perpendicular generators of the cone $5yz-8zx-3xy=0$, then find the equations of the other two. [2026/15m/3c]

3. Find the equation of the right circular cylinder whose axis is $x=2y=-z$, and radius is 4. Prove that the area of the section of this cylinder by the plane $z=0$ is $24\pi$. [2026/10m/4c(ii)]

4. Find the equation of the cone whose vertex is the point $(1,1,0)$ and whose guiding curve is $y=0,x^2+z^2=4$. [2025/10m/1e]

5. Find the equation of the cylinder whose generators are parallel to the line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and that passes through the curve $x^2+y^2=16,z=0$. [2025/10m/2c(i)]

6. Find the equations of the spheres which pass through the circle $x^2+y^2+z^2-2x+2y+4z-3=0,2x+y+z=4$ and touch the plane $3x+4y=14$. [2025/15m/3b]

7. Show that there is no tangent plane to the sphere $x^2+y^2+z^2-4x+2y-4z+4=0$ that can be passed through the straight line $\frac{x+6}{2}=y+3=z+1.$ [2025/15m/4a]

8. Find the equation of the right circular cylinder which passes through the circle $x^2+y^2+z^2=9,x-y+z=3$. [2024/10m/1e]

9. Find the vertex of the cone $4x^2-y^2+2z^2+2xy-3yz+12x-11y+6z+4=0$. [2024/15m/3c]

10. Find the equation of the sphere which touches the plane $3x+2y-z+2=0$ at the point $(1,-2,1)$ and cuts orthogonally the sphere $x^2+y^2+z^2-4x+6y+4=0$. [2024/15m/4c]

11. The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ meets the coordinate axes in $A$, $B$, $C$ respectively. Prove that the equation of the cone generated by the lines drawn from the origin $O$ to meet the circle $ABC$ is $yz(\frac{b}{c}+\frac{c}{b})+zx(\frac{c}{a}+\frac{a}{c})+xy(\frac{b}{a}+\frac{a}{b})=0.$ [2023/10m/2c(ii)]

12. Find the equation of the sphere through the circle $x^2+y^2+z^2-4x-6y+2z-16=0;3x+y+3z-4=0$ in the following two cases. (i) the point $(1,0,-3)$ lies on the sphere. (ii) the given circle is a great circle of the sphere. [2023/15m/3c]

13. A variable plane passes through a fixed point $(a,b,c)$ and meets the axes at points $A$, $B$ and $C$ respectively. Find the locus of the centre of the sphere passing through the points $O$, $A$, $B$ and $C$, $O$ being the origin. [2022/10m/1e]

14. Find the equation of the sphere of smallest possible radius which touches the straight lines : $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$. [2022/15m/3c]

15. If the plane $ux+vy+wz=0$ cuts the cone $a{x}^2+b{y}^2+c{z}^2=0$ in perpendicular generators, then prove that $(b+c)u^2+(c+a)v^2+(a+b)w^2=0$. [2022/15m/4c]

16. Find the equation of the cylinder whose generators are parallel to the line $x=-\frac{y}{2}=\frac{z}{3}$ and whose guiding curve is $x^2+2y^2=1,z=0$. [2021/10m/1e]

17. Show that the planes, which cut the cone $a{x}^2+b{y}^2+c{z}^2=0$ in perpendicular generators, touch the cone $\frac{x^2}{b+c}+\frac{y^2}{c+a}+\frac{z^2}{a+b}=0$. [2021/20m/2a]

18. A sphere of constant radius $r$ passes through the origin $O$ and cuts the axes at the points $A$, $B$ and $C$. Find, the locus of the foot of the perpendicular drawn from $O$ to the plane $ABC$. [2021/15m/3b]

19. Find the equation of the cylinder whose generators are parallel to the line $\frac{x}{1}=\frac{y}{-2}=\frac{z}{3}$ and whose guiding curve is $x^2+y^2=4,z=2$. [2020/15m/2c]

20. If the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ represents one of a set of three mutually perpendicular generators of the cone $5yz-8zx-3xy=0$, then find the equations of the other two generators. [2020/15m/3c]

21. The plane $x+2y+3z=12$ cuts the axes of coordinates in $A$, $B$, $C$. Find the equations of the circle circumscribing the triangle $ABC$. [2019/10m/2c(i)]

22. Prove that the plane $z=0$ cuts the enveloping cone of the sphere $x^2+y^2+z^2=11$ which has the vertex at $(2,4,1)$ in a rectangular hyperbola. [2019/10m/2c(ii)]

23. Find the equation of the sphere in $xyz$-plane passing through the points $(0,0,0)$, $(0,1,-1)$, $(-1,2,0)$ and $(1,2,3)$. [2018/12m/3d]

24. Find the equation of the cone with $(0,0,1)$ as the vertex and $2x^2-y^2=4,z=0$ as the guiding curve. [2018/13m/4c]

25. A plane passes through a fixed point $(a,b,c)$ and cuts the axes at the points $A$, $B$, $C$ respectively. Find the locus of the centre of the sphere which passes through the origin $O$ and $A$, $B$, $C$. [2017/15m/2b]

26. Show that the plane $2x-2y+z+12=0$ touches the sphere $x^2+y^2+z^2-2x-4y+2z-3=0$. Find the point of contact. [2017/10m/2c]

27. Find the equation of the sphere which passes through the circle $x^2+y^2=4;z=0$ and is cut by the plane $x+2y+2z=0$ in a circle of radius 3. [2016/10m/1d]

28. Show that the cone $3yz-2zx-2xy=0$ has an infinite set of three mutually perpendicular generators. If $\frac{x}{1}=\frac{y}{1}=\frac{z}{2}$ is a generator belonging to one such set, find the other two. [2016/10m/4b]

29. For what positive value of $a$, the plane $ax-2y+z+12=0$ touches the sphere $x^2+y^2+z^2-2x-4y+2z-3=0$ and hence find the point of contact. [2015/10m/1e]

30. If $6x=3y=2z$ represents one of the three mutually perpendicular generators of the cone $5yz-8zx-3xy=0$ then obtain the equations of the other two generators. [2015/13m/2d]

31. Examine whether the plane $x+y+z=0$ cuts the cone $yz+zx+xy=0$ in perpendicular lines. [2014/10m/1e]

32. Find the co-ordinates of the points on the sphere $x^2+y^2+z^2-4x+2y=4$, the tangent planes at which are parallel to the plane $2x-y+2z=1$. [2014/10m/4a(i)]

33. Prove that the equation $a{x}^2+b{y}^2+c{z}^2+2ux+2vy+2wz+d=0$, represents a cone if $\frac{u^2}{a}+\frac{v^2}{b}+\frac{w^2}{c}=d$. [2014/10m/4a(ii)]

34. A sphere $S$ has points $(0,1,0)$, $(3,-5,2)$ at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere $S$ with the plane $5x-2y+4z+7=0$ as a great circle. [2013/10m/1e]

35. Show that three mutually perpendicular tangent lines can be drawn to the sphere $x^2+y^2+z^2=r^2$ from any point on the sphere $2(x^2+y^2+z^2)=3r^2$. [2013/15m/4a]

36. A cone has for its guiding curve the circle $x^2+y^2+2ax+2by=0,z=0$ and passes through a fixed point $(0,0,c)$. If the section of the cone by the plane $y=0$ is a rectangular hyperbola, prove that the vertex lies on the fixed circle $\begin{array}{cc}\hfill x^2+y^2+z^2+2ax+2by\hfill & \hfill =0\hfill \\ \hfill 2ax+2by+cz\hfill & \hfill =0.\hfill \end{array}$ [2013/15m/4b]

37. A variable plane is parallel to the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=0$ and meets the axes in $A$, $B$, $C$ respectively. Prove that the circle $ABC$ lies on the cone $yz(\frac{b}{c}+\frac{c}{b})+zx(\frac{c}{a}+\frac{a}{c})+xy(\frac{a}{b}+\frac{b}{a})=0$ [2012/20m/4b]

38. Show that the equation of the sphere which touches the sphere $4(x^2+y^2+z^2)+10x-25y-2z=0$ at the point $(1,2,-2)$ and passes through the point $(-1,0,0)$ is $x^2+y^2+z^2+2x-6y+1=0.$ [2011/10m/1f]

39. Show that the cone $yz+zx+xy=0$ cuts the sphere $x^2+y^2+z^2=a^2$ in two equal circles, and find their area. [2011/20m/4b]

40. Show that the plane $x+y-2z=3$ cuts the sphere $x^2+y^2+z^2-x+y=2$ in a circle of radius 1 and find the equation of the sphere which has this circle as a great circle. [2010/12m/1e]

41. Show that every sphere through the circle $x^2+y^2-2ax+r^2=0,z=0$ cuts orthogonally every sphere through the circle $x^2+z^2=r^2,y=0$ [2010/20m/3c]

42. Find the equation of the sphere having its centre on the plane $4x-5y-z=3$, and passing through the circle $\begin{array}{cc}\hfill x^2+y^2+z^2-12x-3y+4z+8\hfill & \hfill =0\hfill \\ \hfill 3x+4y-5z+3\hfill & \hfill =0\hfill \end{array}$ [2009/12m/1f]

43. Find the equations (in symmetric form) of the tangent line to the sphere $x^2+y^2+z^2+5x-7y+2z-8=0$, $3x-2y+4z+3=0$ at the point $(-3,5,4)$. [2008/12m/1f]

44. A sphere $S$ has points $(0,1,0)$, $(3,-5,2)$ at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere $S$ with the plane $5x-2y+4z+7=0$ as a great circle. [2008/20m/2c]

45. If $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ represent one of a set of three mutually perpendicular generators of the cone $5yz-8zx-3xy=0,$ find the equations of the other two. [2008/20m/3c]

46. Show that the enveloping cylinders of the ellipsoid $a{x}^2+b{y}^2+c{z}^2=1$ with generators perpendicular to $z$ axis meet the plane $z=0$ in parabolas. [2008/20m/4c]

3.3] Conicoids and second degree equations

22 questions

1. Find the equations of the generating lines of the hyperboloid $\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{16}=1$ which pass through the points $(2,3,-4)$ and $(2,-1,\frac{4}{3})$. [2026/10m/4c(i)]

2. Show that the equation $2x^2+3y^2-8x+6y-12z+11=0$ represents an elliptic paraboloid. Also find its principal axis and principal planes. [2023/10m/2c(i)]

3. If $P$, $Q$, $R$; $P\prime$, $Q\prime$, $R\prime$ are feet of the six normals drawn from a point to the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$, and the plane $PQR$ is represented by $lx+my+nz=p$, show that the plane $P\prime Q\prime R\prime$ is given by $\frac{x}{a^{2}l}+\frac{y}{b^{2}m}+\frac{z}{c^{2}n}+\frac{1}{p}=0$. [2022/20m/2c]

4. Find the equations of the tangent plane to the ellipsoid $2x^2+6y^2+3z^2=27$ which passes through the line $x-y-z=0=x-y+2z-9$. [2020/10m/1e]

5. Find the locus of the point of intersection of the perpendicular generators of the hyperbolic paraboloid $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2z$. [2020/15m/4b]

6. Prove that, in general, three normals can be drawn from a given point to the paraboloid $x^2+y^2=2az$, but if the point lies on the surface $27a(x^2+y^2)+8{(a-z)}^3=0$ then two of the three normals coincide. [2019/15m/3b]

7. Find the length of the normal chord through a point $P$ of the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and prove that if it is equal to $4P{G}_3$, where $G_3$ is the point where the normal chord through $P$ meets the $xy$-plane, then $P$ lies on the cone $\frac{x^2}{a^6}(2c^2-a^2)+\frac{y^2}{b^6}(2c^2-b^2)+\frac{z^2}{c^4}=0$ [2019/15m/4b]

8. Find the equations to the generating lines of the paraboloid $(x+y+z)(2x+y-z)=6z$ which pass through the point $(1,1,1)$. [2018/13m/3c]

9. Find the equation of the tangent plane at point $(1,1,1)$ to the conicoid $3x^2-y^2=2z$. [2017/10m/1d]

10. Find the locus of the point of intersection of three mutually perpendicular tangent planes to $a{x}^2+b{y}^2+c{z}^2=1$. [2017/10m/3d]

11. Reduce the following equation to the standard form and hence determine the nature of the conicoid : $x^2+y^2+z^2-yz-zx-xy-3x-6y-9z+21=0$. [2017/15m/4a]

12. Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid $a{x}^2+b{y}^2+c{z}^2=1$. [2016/15m/4d]

13. Two perpendicular tangent planes to the paraboloid $x^2+y^2=2z$ intersect in a straight line in the plane $x=0$. Obtain the curve to which this straight line touches. [2015/13m/4c]

14. Show that the lines drawn from the origin parallel to the normals to the central conicoid $a{x}^2+b{y}^2+c{z}^2=1$, at its points of intersection with the plane $lx+my+nz=p$ generate the cone $p^2(\frac{x^2}{a}+\frac{y^2}{b}+\frac{z^2}{c})={(\frac{lx}{a}+\frac{my}{b}+\frac{nz}{c})}^2.$ [2014/15m/4b]

15. Find the equations of the two generating lines through any point $(acos\theta ,bsin\theta ,0)$, of the principal elliptic section $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $z=0$, of the hyperboloid by the plane $z=0$. [2014/15m/4c]

16. A variable generator meets two generators of the system through the extremities $B$ and $B\prime$ of the minor axis of the principal elliptic section of the hyperboloid $\frac{x^2}{a^2}+\frac{y^2}{b^2}-z^2{c}^2=1$ in $P$ and $P\prime$. Prove that $BP\cdot B\prime P\prime =a^2+c^2$. [2013/20m/4c]

17. Show that the locus of a point from which the three mutually perpendicular tangent lines can be drawn to the paraboloid $x^2+y^2+2z=0$ is $x^2+y^2+4z=1$ [2012/20m/4c]

18. Three points $P$, $Q$, $R$ are taken on the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ so that the lines joining $P$, $Q$, $R$ to the origin are mutually perpendicular. Prove that the plane $PQR$ touches a fixed sphere. [2011/20m/4a]

19. Show that the generators through any one of the ends of an equiconjugate diameter of the principal elliptic section of the hyperboloid $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$ are inclined to each other at an angle of ${60}^{\circ }$ if $a^2+b^2=6c^2$. Find also the condition for the generators to be perpendicular to each other. [2011/20m/4c]

20. Show that the plane $3x+4y+7z+\frac{5}{2}=0$ touches the paraboloid $3x^2+4y^2=10z$ and find the point of contact. [2010/20m/2c]

21. Find the vertices of the skew quadrilateral formed by the four generators of the hyperboloid $\frac{x^2}{4}+y^2-z^2=49$ passing through $(10,5,1)$ and $(14,2,-2)$. [2010/20m/4c]

22. Prove that the normals from the point $(\alpha ,\beta ,\gamma )$ to the paraboloid $\frac{x^2}{a^2}+\frac{y^2}{b^2}=2z$ lie on the cone $\frac{\alpha }{x-\alpha }+\frac{\beta }{y-\beta }+\frac{a^2-b^2}{z-\gamma }=0$ [2009/20m/4c]

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Last updated: 30 October 2026