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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.

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Questions 1-50 of 210

  1. The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S$ ($>R$) respectively from the origin, is :
  2. The tangents, to the parabola $y^{2}=2-x$ at the points of its intersection with the line $y=x-2$, intersect at the point
  3. Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin, respectively. If tangents are drawn …
  4. The locus of the mid-points of the perpendiculars to the directrix of the parabola, $y^{2}=2x$ from points on it, is
  5. Let $O$ be the vertex of the parabola $y^{2}=4x$ and its chords $OP$ and $OQ$ are perpendicular to each other. If the locus of the mid-point of the line segment $PQ$ is …
  6. A tangent and a normal are drawn at the point $P(2,-4)$ on the parabola $y^{2}=8x$, which meet the directrix of the parabola at the points $A$ and $B$ respectively. If …
  7. If two tangents drawn from a point $P$ to the parabola $y^{2}=16(x-3)$ are at right angles, then the locus of point $P$ is :
  8. Let $C$ be the locus of the mirror image of a point on the parabola $y^{2}=4x$ with respect to the line $y=x$. Then the equation of tangent to $C$ at $P(2,1)$ is :
  9. Let $PQ$ be a focal chord of the parabola $y^{2}=36x$ of length $100$, making an acute angle with the positive $x$-axis. Let the ordinate of $P$ be positive and $M$ be …
  10. Let the directrix of the parabola $P:y^{2}=8x$, cut $x$-axis at the point $A$. Let $B(\alpha,\beta)$, $\alpha>1$, be a point on $P$ such that the slope of $AB$ is $3/5$. …
  11. Let $R$ be the focus of the parabola $y^{2}=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10,10)$ be the centroid of the …
  12. Let a line $L:2x+y=k$, $k>0$ be a tangent to the hyperbola $x^{2}-y^{2}=3$. If $L$ is also a tangent to the parabola $y^{2}=\alpha x$, then $\alpha$ is equal to :
  13. Let $A(0,1)$, $B(1,1)$ and $C(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y^{2}=4ax$ passing through …
  14. The locus of the mid-point of the line segment joining the focus of the parabola $y^{2}=4ax$ to a moving point of the parabola, is another parabola whose directrix is :
  15. Let the parabola $y=x^{2}+px+q$ passing through the point $(1,-1)$ be such that the distance between its vertex and the $x$-axis is minimum. Then the value of …
  16. A tangent is drawn to the parabola $y^{2}=6x$ which is perpendicular to the line $2x+y=1$. Which of the following points does NOT lie on it ?
  17. The shortest distance between the line $x-y=1$ and the curve $x^{2}=2y$ is :
  18. If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y=4x-1$, then the co-ordinates of $P$ are :
  19. The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the ratio $3:1$. If $R(\alpha,\beta)$ is the point of intersection …
  20. Let $P(a,b)$ be a point on the parabola $y^2=8x$ such that the tangent at $P$ passes through the centre of the circle $x^2+y^2-10x-14y+65=0$. Let $A$ be the product of …
  21. Let $P:y^{2}=4ax$, $a>0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ make an angle of $\frac{\pi}{4}$ with the line $y=3x+5$ touch the parabola …
  22. Consider the circle $C:x^{2}+y^{2}=4$ and the parabola $P:y^{2}=8x$. If the set of all values of $\alpha$, for which three chords of the circle $C$ on three distinct …
  23. The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^2=4y$, internally in the ratio $1:2$, is :
  24. Let $x=2t$, $y=\frac{t^{2}}{3}$ be a conic. Let $S$ be the focus and $B$ be the point on the axis of the conic such that $SA\perp BA$, where $A$ is any point on the …
  25. Let $L_1,L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9x^{2}+12x+18y-14=0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ …
  26. If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3x+y-29=0$, is $x^{2}+ay^{2}+bxy+cx+dy+k=0$, then $a+b+c+d+k$ is equal to
  27. Let $A,B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^{2}=16x$. Let the vertex $B$ containing the right angle be $(4,8)$ and …
  28. The shortest distance between the curves $y^{2}=8x$ and $x^{2}+y^{2}+12y+35=0$ is:
  29. If the line $y=4+kx$, $k>0$, is the tangent to the parabola $y=x-x^{2}$ at the point $P$ and $V$ is the vertex of the parabola, then the slope of the line through $P$ …
  30. A circle of radius $2$ unit passes through the vertex and the focus of the parabola $y^{2}=2x$ and touches the parabola $y=\left(x-\frac{1}{4}\right)^{2}+\alpha$, where …
  31. Let a line $y=mx$ ($m>0$) intersect the parabola, $y^2=x$ at a point $P$, other than the origin. Let the tangent to it at $P$ meet the $x$-axis at the point $Q$. If area …
  32. Let the tangent to the curve $x^{2}+2x-4y+9=0$ at the point $P(1,3)$ on it meet the $y$-axis at $A$. Let the line passing through $P$ and parallel to the line $x-3y=6$ …
  33. Two tangent lines $l_1$ and $l_2$ are drawn from the point $(2,0)$ to the parabola $2y^2=-x$. If the lines $l_1$ and $l_2$ are also tangent to the circle …
  34. Let the curves $y^2=kx$ and $xy=-1$ have a common tangent, whose slope is $\frac12$. Then $k$ can NOT lie in the interval
  35. Let $A$, $B$ and $C$ be three points on the parabola $y^{2}=6x$ and let the line segment $AB$ meet the line $L$ through $C$ parallel to the $x$-axis at the point $D$. …
  36. Let $P$ be a point on the parabola, $y^2=12x$ and $N$ be the foot of the perpendicular drawn from $P$ on the axis of the parabola. A line is now drawn through the …
  37. If the shortest distance of the parabola $y^{2}=4x$ from the centre of the circle $x^{2}+y^{2}-4x-16y+64=0$ is $d$, then $d^{2}$ is equal to :
  38. Let the line $L:\sqrt{2}x+y=\alpha$ pass through the point of the intersection $P$ (in the first quadrant) of the circle $x^{2}+y^{2}=3$ and the parabola $x^{2}=2y$. Let …
  39. If $y=m_1x+c_1$ and $y=m_2x+c_2$, $m_1\neq m_2$ are two common tangents of circle $x^{2}+y^{2}=2$ and parabola $y^{2}=x$, then the value of $8|m_1m_2|$ is equal to :
  40. Let the function $f(x)=2x^2-\log_ex$, $x>0$, be decreasing in $(0,a)$ and increasing in $(a,4)$. A tangent to the parabola $y^2=4ax$ at a point $P$ on it passes through …
  41. Let $P$ be the parabola, whose focus is $(-2,1)$ and directrix is $2x+y+2=0$. Then the sum of the ordinates of the points on $P$, whose abscissa is $-2$, is
  42. The equation of a common tangent to the parabolas $y=x^2$ and $y=-(x-2)^2$ is
  43. If the tangents drawn at the points $P$ and $Q$ on the parabola $y^2=2x-3$ intersect at the point $R(0,1)$, then the orthocentre of the triangle $PQR$ is :
  44. The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin, respectively. …
  45. If the length of the latus rectum of a parabola, whose focus is $(a,a)$ and the tangent at its vertex is $x+y=a$, is $16$, then $|a|$ is equal to :
  46. If one end of a focal chord $AB$ of the parabola $y^2=8x$ is at $A\left(\frac12,-2\right)$, then the equation of the tangent to it at $B$ is :
  47. The radius of the smallest circle which touches the parabolas $y=x^{2}+2$ and $x=y^{2}+2$ is
  48. Let the focal chord $PQ$ of the parabola $y^{2}=4x$ make an angle of $60^{\circ}$ with the positive $x$-axis, where $P$ lies in the first quadrant. If the circle, whose …
  49. If two tangents drawn from a point $(\alpha,\beta)$ lying on the ellipse $25x^{2}+4y^{2}=1$ to the parabola $y^{2}=4x$ are such that the slope of one tangent is four …
  50. The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^2=8x$, with one of its vertices on the vertex of this parabola, is :

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