iCON Education - Parabola AP-EAPCET Test

Question 1: Palette 1/27

AP-EAPCET 2025 MCQ

If $x-y-3=0$ is a normal drawn through the point $(5,2)$ to the parabola $y^2=4 x$, then the slope of the other normal that can be drawn through the same point to the parabola $y^2=4 x$ is

-1

-2

Question 2: Palette 2/27

AP-EAPCET 2025 MCQ

A circle is drawn with its centre at the focus of the parabola $y^2=2 p x$ such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is

$(2 p, 2 p)$

$\left(\frac{p}{2},-p\right)$

$(2 p,-2 p)$

$(p, \sqrt{2} p)$

Question 3: Palette 3/27

AP-EAPCET 2025 MCQ

If the locus of a point that divides a chord of slope 2 of the parabola $y^2=4 x$ internally in the ratio $1: 2$ is a parabola, then its vertex is

$\left(\frac{2}{9}, \frac{8}{9}\right)$

$\left(\frac{1}{9}, \frac{3}{9}\right)$

$\left(\frac{4}{9}, \frac{8}{9}\right)$

$\left(\frac{2}{9}, \frac{4}{9}\right)$

Question 4: Palette 4/27

AP-EAPCET 2025 MCQ

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15 x$ subtends an angle $\theta$ at the vertex of the parabola, then $\sin \frac{\theta}{3}+\cos \frac{2 \theta}{3}-\sec \frac{4 \theta}{3}=$

Question 5: Palette 5/27

AP-EAPCET 2025 MCQ

Tangents are drawn at three points $P\left(t_1\right), Q\left(t_2\right), R\left(t_3\right)$ on the parabola $y^2=x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1=2, t_2=-4, t_3=6$, then the area of the $\triangle L M N$ is

18.5

7.5

Question 6: Palette 6/27

AP-EAPCET 2025 MCQ

If the tangents of the parabola $y^2=8 x$ passing through the point $P(1,3)$ touches the parabola at $A$ and $B$, then the area (in sq. units) of $\triangle P A B$ is

$\frac{3}{4}$

$\frac{1}{2}$

$\frac{1}{4}$

Question 7: Palette 7/27

AP-EAPCET 2025 MCQ

The lengths of the two focal chords of the parabola $y^2=16 x$ is 25 units each. If these two chords cut the parabola at $A, B, C$ and $D$, then the area (in sq. units) of the quadrilateral formed by $A, B, C$ and $D$ is

$\frac{625}{2}$

180

150

300

Question 8: Palette 8/27

AP-EAPCET 2025 MCQ

If the perpendicular distance from the focus of a parabola $y^2=4 a x$ to its directrix is $\frac{3}{2}$, then the equation of the normal drawn at $(4 a,-4 a)$ is

$2 x+y=3$

$2 x-y=9$

$x-2 y=9$

$x+2 y+3=0$

Question 9: Palette 9/27

AP-EAPCET 2025 MCQ

$P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$. If $P=(4,4)$, then $S Q=$

$\frac{5}{4}$

$\frac{3}{2}$

Question 10: Palette 10/27

AP-EAPCET 2025 MCQ

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4 x$ is

$\frac{\pi}{6}$

$\frac{\pi}{4}$

$\frac{\pi}{3}$

$\frac{\pi}{2}$

Question 11: Palette 11/27

AP-EAPCET 2025 MCQ

If $L$ is the normal drawn to the parabola $y^2=8 x$ at the point $t=\frac{1}{\sqrt{2}}$, then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

$(3,2)$

$(5, \sqrt{2})$

$(0, \sqrt{2})$

$(3, \sqrt{2})$

Question 12: Palette 12/27

AP-EAPCET 2024 MCQ

If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12 x$ and a point on the parabola in the ratio $1: 2$. Then, the locus of $p$ is

$y^2=2(x-2)$

$y^2=4 x$

$y^2=4(x-2)$

$y^2=9(x-3)$

Question 13: Palette 13/27

AP-EAPCET 2024 MCQ

Equation of the line touching both parabolas $y^2=4 x$ and $x^2=-32 y$ is

$x+2 y+4=0$

$2 x+y-4=0$

$x-2 y-4=0$

$x-2 y+4=0$

Question 14: Palette 14/27

AP-EAPCET 2024 MCQ

If the normal chord drawn at $(2 a, 2 a \sqrt{2})$ on the parabola $y^2=4 a x$ subtends an angle $\theta$ at its vertex, then $\theta=$

$45^{\circ}$

$90^{\circ}$

$135^{\circ}$

$60^{\circ}$

Question 15: Palette 15/27

AP-EAPCET 2024 MCQ

If the ordinates of points $P$ and $Q$ on the parabola $y^2=12 x$ are in the ratio $1: 2$. Then, the locus of the point of intersection of the normals to the parabola at $P$ and $Q$ is

$y+18\left(\frac{x-6}{21}\right)^{\frac{3}{2}}=0$

$y-18\left(\frac{x-6}{12}\right)^{\frac{3}{2}}=0$

$y+12\left(\frac{x-6}{14}\right)^{\frac{1}{2}}=0$

$y-12\left(\frac{x-6}{18}\right)^{\frac{1}{2}}=0$

Question 16: Palette 16/27

AP-EAPCET 2024 MCQ

A common tangent to the circle $x^2+y^2=9$ and parabola $y^2=8 x$ is

$3 x-\sqrt{3 y}+2=0$

$x-\sqrt{3} y+6=0$

$2 x-\sqrt{3} y+3=0$

$x-3 y+6=0$

Question 17: Palette 17/27

AP-EAPCET 2024 MCQ

The normal drawn at a point $(2,-4)$ on the parabola $y^2 \pm 8 x$ cuts again the same parabola at $(\alpha, \beta)$, then $\alpha+\beta=$

Question 18: Palette 18/27

AP-EAPCET 2024 MCQ

If the axes are rotated through an angle $45^{\circ}$ about the origin in anticlockwise direction, then the transformed equation of $y^2=4 a r$ is

$(x+y)^2=4 \sqrt{2} a(x-y)$

$(x-y)^2=4 \sqrt{2} a(x+y)$

$(x-y)^2=\frac{43}{\sqrt{2}}(x-y)$

$(x+y)^2=\frac{4 a}{\sqrt{2}}(x-y)$

Question 19: Palette 19/27

AP-EAPCET 2024 MCQ

The line $x-2 y-3=0$ cuts the parabola $y^2=4 \operatorname{ar}$ at the points $P$ and $Q$. If the focus of this parabola is $\left(\frac{1}{4}, k\right)$. then $P Q=$

$16 a \sqrt{5}$

$8 a \sqrt{5}$

$4 a \sqrt{5}$

$2 a \sqrt{5}$

Question 20: Palette 20/27

AP-EAPCET 2022 MCQ

Which of the following represents a parabola?

$x=4 \cos t, y=4 \sin t$

$x^2-2=-2 \cos t, y=\cos ^2\left(\frac{t}{2}\right)$

$\sqrt{x}=\tan t, \sqrt{y}=\sec t$

$x=\sqrt{1-\sin t}, y=\sin \left(\frac{t}{2}\right)+\cos \left(\frac{t}{2}\right)$

Question 21: Palette 21/27

AP-EAPCET 2022 MCQ

Suppose a parabola passes through $(0,4),(1,9)$ and $(4,5)$ and has its axis parallel to the $Y$-axis. Then, the equation of the parabola is

$19 x^2+12 y-79 x-48=0$

$19 x^2+12 y-79 x+48=0$

$19 y^2+12 x-79 y-48=0$

$19 y^2+12 x-79 y+48=0$

Question 22: Palette 22/27

AP-EAPCET 2022 MCQ

Suppose a parabola with focus at $(0,0)$ has $x-y+1=0$ as its tangent at the vertex. Then, the equation of its directrix is

$x-y+2=0$

$x-y-2=0$

$x-y+3=0$

$x-y+4=0$

Question 23: Palette 23/27

AP-EAPCET 2022 MCQ

If $a x+b y=1$ is a normal to the parabola $y^2=4 p x$, then the condition is

$4 a b=a^2+b^2$

$4 p a b+a b^3=a^2 b^2$

$p a^3=b^2-2 p a b^2$

$p a^2+4 p a=a+b$

Question 24: Palette 24/27

AP-EAPCET 2021 MCQ

The point of intersection of the latus rectum and axis of the parabola $y^2+4 x+2 y-8=0$ is

$\left(\frac{9}{4},-1\right)$

$\left(\frac{5}{4},-1\right)$

$\left(\frac{7}{2}, \frac{5}{2}\right)$

$\left(\frac{-5}{4}, 1\right)$

Question 25: Palette 25/27

AP-EAPCET 2021 MCQ

The coordinates of the focus of the parabola described parametrically by $x=5t^2+2$ and $y=10t+4$ (where t is a parameter) are

(7, 4)

(3, 4)

(3, $-$4)

($-$7, 4)

Question 26: Palette 26/27

AP-EAPCET 2021 MCQ

Find the equation of the parabola which passes through (6, $-$2), has its vertex at the origin and its axis along the Y-axis.

$y^2=18x$

$x^2=18y$

$y^2=-18x$

$x^2=-18y$

Question 27: Palette 27/27

AP-EAPCET 2021 MCQ

If one end of focal chord of the parabola $y^2=8x$ is $\left(\frac{1}{2},2\right)$, then the length of the focal chord is ................ units.

$\frac{625}{4}$

$\frac{5}{\sqrt2}$

$\frac{25}{2}$