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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.

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Questions 51-100 of 210

  1. Let $PQ$ be a focal chord of the parabola $y^{2}=4x$ such that it subtends an angle of $\frac{\pi}{2}$ at the point $(3,0)$. Let the line segment $PQ$ be also a focal …
  2. The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2-2x-2y=1$ meet at the point $P$. Then the area (in unit$^2$) of the triangle $PAB$ is:
  3. If vertex of a parabola is $(2,-1)$ and the equation of its directrix is $4x-3y=21$, then the length of its latus rectum is :
  4. The sum of diameters of the circles that touch (i) the parabola $75x^2=64(5y-3)$ at the point $\left(\frac85,\frac65\right)$ and (ii) the $y$-axis, is equal to ________.
  5. Let a common tangent to the curves $y^{2}=4x$ and $(x-4)^{2}+y^{2}=16$ touch the curves at the points $P$ and $Q$. Then $(PQ)^{2}$ is equal to $\_\_\_\_\_\_$ .
  6. A particle is moving in the $xy$-plane along a curve $C$ passing through the point $(3,3)$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If …
  7. Let $P_1$ be a parabola with vertex $(3,2)$ and focus $(4,4)$ and $P_2$ be its mirror image with respect to the line $x+2y=6$. Then the directrix of $P_2$ is …
  8. Let the point $P$ of the focal chord $PQ$ of the parabola $y^{2}=16x$ be $(1,-4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m:n$, $\gcd(m,n)=1$ …
  9. Let a conic $C$ pass through the point $(4,-2)$ and $P(x,y)$, $x\ge3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ …
  10. The equation of a circle is $\operatorname{Re}\left(z^{2}\right)+2(\operatorname{Im}(z))^{2}+2\operatorname{Re}(z)=0$, where $z=x+iy$. A line which passes through the …
  11. The solution curve, of the differential equation $2y\frac{dy}{dx}+3=5\frac{dy}{dx}$, passing through the point $(0,1)$ is a conic, whose vertex lies on the line :
  12. Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the $x$-intercept of the normal to the curve at the …
  13. Let $PQ$ be a focal chord of length $6.25$ units of the parabola $y^{2}=4x$. If $O$ is the vertex of the parabola, then $10$ times the area (in sq. units) of …
  14. Let the tangent drawn to the parabola $y^2=24x$ at the point $(\alpha,\beta)$ is perpendicular to the line $2x+2y=5$. Then the normal to the hyperbola …
  15. Let $r$ be the radius of the circle, which touches $x$-axis at point $(a,0)$, $a<0$ and the parabola $y^{2}=9x$ at the point $(4,6)$. Then $r$ is equal to $\_\_\_\_\_\_$ …
  16. Let the normals at all the points on a given curve pass through a fixed point $(a,b)$. If the curve passes through $(3,-3)$ and $(4,-2\sqrt{2})$, and given that …
  17. Let the latus rectum of the parabola $y^2=4x$ be the common chord to the circles $C_1$ and $C_2$ each of them having radius $2\sqrt5$. Then, the distance between the …
  18. Let the normal at the point $P$ on the parabola $y^{2}=6x$ pass through the point $(5,-8)$. If the tangent at $P$ to the parabola intersects its directrix at the point …
  19. Let $O$ be the vertex and $Q$ be any point on the parabola, $x^{2}=8y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1:3$, then locus of $P$ …
  20. Statement-1: The line $x-2y=2$ meets the parabola, $y^{2}+2x=0$ only at the point $(-2,-2)$. Statement-2: The line $y=mx-\frac{1}{2m}$ $(m\neq0)$ is tangent to the …
  21. Two tangents are drawn from a point $(-2,-1)$ to the curve $y^{2}=4x$. If $\alpha$ is the angle between them, then $|\tan\alpha|$ is equal to :
  22. If $a\neq0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^{2}=4ax$ and $x^{2}=4ay$, then :
  23. A chord is drawn through the focus of the parabola $y^{2}=6x$ such that its distance from the vertex of this parabola is $\frac{\sqrt5}{2}$, then its slope can be
  24. Let $P$ be the point $(1,0)$ and $Q$ a point on the parabola $y^{2}=8x$. The locus of mid point of $PQ$ is :
  25. If two tangents drawn from a point $P$ to the parabola $y^{2}=4x$ are at right angles, then the locus of $P$ is
  26. The tangent to the parabola $y^2=4x$ at the point where it intersects the circle $x^2+y^2=5$ in the first quadrant, passes through the point :
  27. $P$ and $Q$ are two distinct points on the parabola, $y^{2}=4x$, with parameters $t$ and $t_1$, respectively. If the normal at $P$ passes through $Q$, then the minimum …
  28. If the common tangents to the parabola $x^{2}=4y$ and the circle $x^{2}+y^{2}=4$ intersect at the point $P$, then the distance of $P$ from the origin (units), is :
  29. A point on the parabola $y^{2}=18x$ at which the ordinate increases at twice the rate of the abscissa is
  30. If the tangent to the parabola $y^{2}=x$ at a point $(\alpha,\beta)$, $(\beta>0)$ is also a tangent to the ellipse, $x^{2}+2y^{2}=1$, then $\alpha$ is equal to :
  31. A parabola has the origin as its focus and the line $x=2$ as the directrix. Then the vertex of the parabola is at :
  32. The locus of the vertices of the family of parabolas $y=\frac{a^{3}x^{2}}{3}+\frac{a^{2}x}{2}-2a$ is :
  33. If $y=mx+c$ is the normal at a point on the parabola $y^{2}=8x$ whose focal distance is $8$ units, then $|c|$ is equal to :
  34. The point of intersection of the normals to the parabola $y^{2}=4x$ at the ends of its latus rectum is :
  35. Let $L_1$ be the length of the common chord of the curves $x^2+y^2=9$ and $y^2=8x$, and $L_2$ be the length of the latus rectum of $y^2=8x$, then :
  36. If the tangent at a point $P$ on the parabola $y^{2}=3x$ is parallel to the line $x+2y=1$ and the tangents at the points $Q$ and $R$ on the ellipse …
  37. Let $S$ be the set of all $a\in N$ such that the area of the triangle formed by the tangent at the point $P(b,c)$, $b,c\in N$, on the parabola $y^{2}=2ax$ and the lines …
  38. If $PQ$ be a double ordinate of the parabola $y^{2}=-4x$, where $P$ lies in the second quadrant. If $R$ divides $PQ$ in the ratio $2:1$, then the locus of $R$ is :
  39. The distance of the point $\left(6,-2\sqrt2\right)$ from the common tangent $y=mx+c,\ m>0$, of the curves $x=2y^{2}$ and $x=1+y^{2}$ is
  40. If one end of a focal chord of the parabola $y^{2}=16x$ is at $(1,4)$, then the length of this focal chord is
  41. If $y=mx+4$ is a tangent to both the parabolas, $y^{2}=4x$ and $x^{2}=2by$, then $b$ is equal to
  42. If the area of the triangle whose one vertex is at the vertex of the parabola, $y^{2}+4\left(x-a^{2}\right)=0$ and the other two vertices are the points of intersection …
  43. A circle cuts a chord of length $4a$ on the $x$-axis and passes through a point on the $y$-axis, distant $2b$ from the origin. Then the locus of the centre of this …
  44. The length of the chord of the parabola $x^2=4y$ having equation $x-\sqrt2y+4\sqrt2=0$ is :
  45. Let the tangent to the parabola $S:y^{2}=2x$ at the point $P(2,2)$ meet the $x$-axis at $Q$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in …
  46. The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^2=6x$. The locus of its circumcentre is :
  47. Let $y^{2}=12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)=\frac{147}4$. Let $C$ be the circle described taking …
  48. Tangent and normal are drawn at $P(16,16)$ on the parabola $y^2=16x$, which intersect the axis of the parabola at $A$ and $B$, respectively. If $C$ is the center of the …
  49. Let $P$ be the point on the parabola $y^{2}=8x$ which is at a minimum distance from the centre $C$ of the circle $x^{2}+(y+6)^{2}=1$. Then the equation of the circle …
  50. Consider the parabola with vertex $\left(\frac12,\frac34\right)$ and the directrix $y=\frac12$. Let $P$ be the point where the parabola meets the line $x=-\frac12$. If …

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