Parabola-JEE-Mains-PYQ’s

Mathematics Questions

Let $O$ be the vertex of the parabola $x^2 = 4y$ and $Q$ be any point on it. Let the locus of the point $P$ , which divides the line segment $OQ$ internally in the ratio $2 : 3$ be the conic $C$ . Then the equation of the chord of $C$ , which is bisected at the point $(1, 2)$ , is :

(A) $5x - 4y + 3 = 0$

(B) $x - 2y + 3 = 0$

(D) $5x - y - 3 = 0$

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If the line $\alpha x + 4y = \sqrt{7}$ , where $\alpha \in \mathbb{R}$ , touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is :

(A) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{11}}$

(B) $\frac{1}{\sqrt{3}} - \frac{1}{2\sqrt{5}}$

(D) $\frac{1}{\sqrt{3}} + \frac{1}{2\sqrt{7}}$

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Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36} + \frac{y^2}{16} = 1$ . If the eccentricity of the hyperbola is $5$ , then the length of its latus rectum is :

(A) $\frac{96}{\sqrt{5}}$

(B) $24\sqrt{5}$

(D) $16$

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Let $A$ be the focus of the parabola $y^{2}=8x$ . Let the line $y=mx+c$ intersect the parabola at two distinct points $B$ and $C$ . If the centroid of the triangle $ABC$ is $\left(\frac{7}{3}, \frac{4}{3}\right)$ , then $(BC)^{2}$ is equal to :

(A) 89

(B) 80

(D) 41

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Let the image of parabola $x^{2}=4y$ , in the line $x-y=1$ be $(y+a)^{2}=b(x-c), a, b, c \in \mathbb{N}$ . Then $a+b+c$ is equal to

(A) 12

(B) 8

(D) 4

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An equilateral triangle $OAB$ is inscribed in the parabola $y^{2}=4x$ with the vertex $O$ at the vertex of the parabola. Then the minimum distance of the circle having $AB$ as a diameter from the origin is

(A) $2(8-3\sqrt{3})$

(B) $4(6+\sqrt{3})$

(D) $2(3+\sqrt{3})$

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Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^{2}=4x$ be the curve $S$ . Let $P$ be any point on $S$ . Then the locus of the point, which internally divides $OP$ in the ratio $3:1$ , is :

(A) $2x^{2}=3y$

(B) $2y^{2}=3x$

(D) $3x^{2}=2y$

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If the chord joining the points $P_{1}(x_{1}, y_{1})$ and $P_{2}(x_{2}, y_{2})$ on the parabola $y^{2}=12x$ subtends a right angle at the vertex of the parabola, then $x_{1}x_{2}-y_{1}y_{2}$ is equal to

(A) 280

(B) 288

(D) 284

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Let $y^{2}=12x$ be the parabola with its vertex at $O$ . Let $P$ be a point on the parabola and $A$ be a point on the $x$ -axis such that $\angle OPA = 90^{\circ}$ . Then the locus of the centroid of such triangles $OPA$ is:

(A) $y^{2}-4x+8=0$

(B) $y^{2}-2x+8=0$

(D) $y^{2}-6x+4=0$

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Let one end of a focal chord of the parabola $y^{2}=16x$ be $(16, 16)$ . If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5:2$ , then the minimum value of $\alpha+\beta$ is equal to:

(A) 5

(B) 7

(D) 16

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Parabola-JEE-Mains-PYQ’s

Mathematics Questions

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