Skip to main content
Yukti Vault

Coordinate Geometry

Parabola: JEE Main & Advanced PYQs

210 previous year questions on Parabola from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

Practice this chapter

Questions 101-150 of 210

  1. Two parabolas with a common vertex and with axes along $x$-axis and $y$-axis, respectively, intersect each other in the first quadrant. If the length of the latus rectum …
  2. Let $T$ be the tangent to the parabola $y^2 = 16x$ at the point $(64, 32)$. Let $L$ be the tangent to the same parabola at another point $(x_1,\ y_1)$ on the parabola. …
  3. Consider the parabola $y^2=4x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P=(-2,\,1)$ meet the parabola at $P_1$ and …
  4. Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola, $y^{2}=4x$ and let $X$ be any point on the arc $POQ$ of this parabola, where $O$ is the vertex of this …
  5. The parabolas : $ax^{2}+2bx+cy=0$ and $dx^{2}+2ex+fy=0$ intersect on the line $y=1$. If $a,b,c,d,e,f$ are positive real numbers and $a,b,c$ are in G.P., then
  6. Tangents drawn from the point $(-8,0)$ to the parabola $y^{2}=8x$ touch the parabola at $P$ and $Q$. If $F$ is the focus of the parabola, then the area of the triangle …
  7. Let $P(\alpha,\beta)$ be a point on the parabola $y^{2}=4x$. If $P$ also lies on the chord of the parabola $x^{2}=8y$ whose mid point is $\left(1,\frac54\right)$, then …
  8. Axis of a parabola lies along $x$-axis. If its vertex and focus are at distances $2$ and $4$ respectively from the origin, on the positive $x$-axis then which of the …
  9. Equation of a common tangent to the circle, $x^2+y^2-6x=0$ and the parabola, $y^2=4x$, is:
  10. If the line $ax+y=c$ touches both the curves $x^{2}+y^{2}=1$ and $y^{2}=4\sqrt2x$, then $|c|$ is equal to :
  11. $A$ is a point on the parabola $y^{2}=4ax$. The normal at $A$ cuts the parabola again at point $B$. If $AB$ subtends a right angle at the vertex of the parabola, find …
  12. Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, $y^{2}=4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of …
  13. Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$ …
  14. The vertices of a triangle are $\left[at_1t_2,\ a(t_1+t_2)\right]$, $\left[at_2t_3,\ a(t_2+t_3)\right]$, $\left[at_3t_1,\ a(t_3+t_1)\right]$. Find the orthocentre of the …
  15. The equation of a tangent to the parabola $x^{2}=8y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is
  16. Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12x$. Suppose that the tangent $T_1$ …
  17. Let $y=mx+c$, $m>0$ be the focal chord of $y^{2}=-64x$, which is tangent to $(x+10)^{2}+y^{2}=4$. Then, the value of $4\sqrt2(m+c)$ is equal to
  18. Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$, such that the area of the region enclosed between the parabola and the chord is …
  19. A normal with slope $\frac{1}{\sqrt6}$ is drawn from the point $(0,-\alpha)$ to the parabola $x^2=-4ay$, where $a>0$. Let $L$ be the line passing through $(0,-\alpha)$ …
  20. If the $x$-intercept of a focal chord of the parabola $y^{2}=8x+4y+4$ is $3$, then the length of this chord is equal to
  21. The equation of a common tangent to the curves, $y^{2}=16x$ and $xy=-4$, is :
  22. If the tangent at a point $P$, with parameter $t$, on the curve $x=4t^{2}+3$, $y=8t^{3}-1$, $t\in R$, meets the curve again at a point $Q$, then the coordinates of $Q$ …
  23. Let $P$ be a point on the parabola $y^2=4ax$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$, where $F$ …
  24. The urns $A$, $B$ and $C$ contain $4$ red, $6$ black; $5$ red, $5$ black and $\lambda$ red, $4$ black balls respectively. One of the urns is selected at random and a …
  25. If $P(h,k)$ be a point on the parabola $x=4y^2$, which is nearest to the point $Q(0,33)$, then the distance of $P$ from the directrix of the parabola $y^2=4(x+y)$ is …
  26. Let $P$ be the point of intersection of the common tangents to the parabola $y^2=12x$ and the hyperbola $8x^2-y^2=8$. If $S$ and $S'$ denote the foci of the hyperbola …
  27. Let $P$ be a point on the parabola $x^{2}=4y$. If the distance of $P$ from the center of the circle $x^{2}+y^{2}+6x+8=0$ is minimum, then the equation of the tangent to …
  28. Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^{2}+y^{2}=8$ and $y^{2}=16x$. If one of these tangents touches the two curves at …
  29. Let $A_1,B_1,C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1C_1$ and $B_1C_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$, respectively. If …
  30. Let a tangent to the curve $y^{2}=24x$ meet the curve $xy=2$ at the points $A$ and $B$. Then the mid-points of such line segments $AB$ lie on a parabola with the
  31. Three normals are drawn from the point $(c,0)$ to the curve $y^{2}=x$. Show that $c$ must be greater than $1/2$. One normal is always the $x$-axis. Find $c$ for which …
  32. Through the vertex $O$ of the parabola $y^{2}=4x$, chords $OP$ and $OQ$ are drawn at right angles to one another. Show that for all positions of $P$, $PQ$ cuts the axis …
  33. Let $C_1$ and $C_2$ be respectively the parabolas $x^{2}=y-1$ and $y^{2}=x-1$. Let $P$ be any point on $C_1$ and $Q$ be any point on $C_2$. Let $P_1$ and $Q_1$ be the …
  34. Consider a circle with its centre lying on the focus of the parabola $y^{2}=2px$ such that it touches the directrix of the parabola. Then a point of intersection of the …
  35. Show that the locus of a point that divides a chord of slope $2$ of the parabola $y^{2}=4x$ internally in the ratio $1:2$ is a parabola. Find the vertex of this parabola.
  36. If $x+y=k$ is normal to $y^{2}=12x$, then $k$ is
  37. The point of intersection of the tangents at the ends of the latus rectum of the parabola $y^{2}=4x$ is ______.
  38. If the line $x-1=0$ is the directrix of the parabola $y^{2}-kx+8=0$, then one of the values of $k$ is
  39. The equation of the directrix of the parabola $y^{2}+4y+4x+2=0$ is
  40. The curve described parametrically by $x=t^{2}+t+1$, $y=t^{2}-t+1$ represents
  41. From a point $A$ common tangents are drawn to the circle $x^{2}+y^{2}=a^{2}/2$ and parabola $y^{2}=4ax$. Find the area of the quadrilateral formed by the common …
  42. The equation of the common tangent touching the circle $(x-3)^{2}+y^{2}=9$ and the parabola $y^{2}=4x$ above the $x$-axis is
  43. Points $A$, $B$ and $C$ lie on the parabola $y^{2}=4ax$. The tangents to the parabola at $A$, $B$ and $C$, taken in pairs, intersect at points $P$, $Q$ and $R$. …
  44. A tangent is drawn to the parabola $y^{2}-2y-4x+5=0$ at a point $P$ which cuts the directrix at the point $Q$. A point $R$ is such that it divides $QP$ externally in the …
  45. The focal chord to $y^2=16x$ is tangent to $(x-6)^2+y^2=2$, then the possible values of the slope of the chord, are
  46. The axis of a parabola is along the line $y=x$ and the distances of its vertex and focus from origin are $\sqrt2$ and $2\sqrt2$ respectively. If vertex and focus both …
  47. Normals are drawn from the point $P$ with slopes $m_1,m_2,m_3$ to the parabola $y^2=4x$. If locus of $P$ with $m_1m_2=\alpha$ is a part of the parabola itself then find …
  48. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2}=4ax$ is another parabola with directrix
  49. The equation of the common tangent to the curves $y^{2}=8x$ and $xy=-1$ is
  50. The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^{2}=4x$ is

Other chapters