JEE Conic Sections PYQs with Video Solutions, Download PDF

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 is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair (a, m) is

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 $$O = (0,0)$$ and $$C_1 = (-4, 0)$$, then which of the following statements is (are) TRUE?

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 $$P_2$$. Let $$Q_1$$ and $$Q_2$$ be points on the lines $$SP_1$$ and $$SP_2$$ respectively such that $$PQ_1$$ is perpendicular to $$SP_1$$ and $$PQ_2$$ is perpendicular to $$SP_2$$. Then, which of the following is/are TRUE?

Consider the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ having one of its focus at P(-3, 0). If the latus rectum through its other focus subtends a right angle at P and $$a^2b^2 = \alpha\sqrt{2} - \beta$$, $$\alpha, \beta \in \mathbb{N}$$.

Tangents are drawn to the hyperbola $$4x^2 - y^2 = 36$$ at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units) of $$\triangle PTQ$$ is:

A stair-case of length $$l$$ rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the staircase begins to slide on the floor, then the locus of P is:

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 the point on the line segment PQ such that $$PM : MQ = 3 : 1$$. Then which of the following points does NOT lie on the line passing through M and perpendicular to the line $$PQ$$?

Consider ellipses $$E_k: kx^2 + k^2y^2 = 1$$, $$k = 1, 2, \ldots, 20$$. Let $$C_k$$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $$E_k$$. If $$r_k$$ is the radius of the circle $$C_k$$, then the value of $$\sum_{k=1}^{20} \frac{1}{r_k^2}$$ is

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 $$x - 3y = 6$$ meet the parabola $$y^2 = 4x$$ at B. If B lies on the line $$2x - 3y = 8$$, then $$AB^2$$ is equal to ______.

Consider the ellipse $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Let $$S(p, q)$$ be a point in the first quadrant such that $$\frac{p^2}{9} + \frac{q^2}{4} \gt 1$$. Two tangents are drawn from $$S$$ to the ellipse, of which one meets the ellipse at one end point of the minor axis and the other meets the ellipse at a point $$T$$ in the fourth quadrant. Let $$R$$ be the vertex of the ellipse with positive $$x$$-coordinate and $$O$$ be the center of the ellipse. If the area of the triangle $$\triangle ORT$$ is $$\frac{3}{2}$$, then which of the following options is correct?

Let the sum of the focal distances of the point $$P(4, 3)$$ on the hyperbola H : $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$8\sqrt{\frac{5}{3}}$$. If for $$H$$, the length of the latus rectum is $$l$$ and the product of the focal distances of the point P is m, then $$9l^2 + 6m$$ is equal to :

For some $$\theta \in \left(0,\frac{\pi}{2}\right)$$, let the eccentricity and the length of the latus rectum of the hyperbola $$x^{2}-y^{2}\sec^{2}\theta =8$$ be $$e_{1}$$ and $$l_{1}$$,respectively, and let the eccentricity and the length of the latus rectum of the ellipse $$x^{2}\sec^{2}\theta +y^{2}=6$$ be $$e_{2}$$ and $$l_{2}$$.respectively. If $$e_{1}^{2}=e_{2}^{2}\left(\sec^{2}\theta +1\right)$$, then $$\left(\frac{l_{1}l_{2}}{e_{1}e_{2}}\right)\tan^{2}\theta$$ is equal to_____

The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:

The equation of a tangent to the parabola, $$x^2 = 8y$$, which makes an angle $$\theta$$ with the positive direction of x-axis, is

Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $$\Delta S'BS$$ is a right angled triangle with right angle at B and area ($$\Delta S'BS$$) = 8 sq. units, then the length of a latus rectum of the ellipse is:

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

Let a tangent to the curve $$9x^2 + 16y^2 = 144$$ intersect the coordinate axes at the points $$A$$ and $$B$$. Then, the minimum length of the line segment $$AB$$ is _____

If the curves $$y^2 = 6x$$, $$9x^2 + by^2 = 16$$ intersect each other at right angles, then the value of b is:

Statement-1: The line $$x - 2y = 2$$ meets the parabola, $$y^2 + 2x = 0$$ only at the point (-2, -2).
Statement-2: The line $$y = mx - \frac{1}{2m}$$ ($$m \neq 0$$) is tangent to the parabola, $$y^2 = -2x$$ at the point $$\left(-\frac{1}{2m^2}, -\frac{1}{m}\right)$$

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:

Equation of a common tangent to the parabola $$y^2 = 4x$$ and the hyperbola $$xy = 2$$ is:

Let $$a$$ and $$b$$ be any two numbers satisfying $$\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{4}$$. Then, the foot of perpendicular from the origin on the variable line $$\frac{x}{a} + \frac{y}{b} = 1$$ lies on:

Tangent and normal are drawn at P(16, 16) on the parabola $$y^2 = 16x$$, which intersect the axis of the parabola at A & B, respectively. If C is the center of the circle through the points P, A & B and $$\angle CPB = \theta$$, then a value of $$\tan \theta$$ is:

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 the parabola at P is:

If the radius of the largest circle with centre (2, 0) inscribed in the ellipse $$x^2 + 4y^2 = 36$$ is $$r$$, then $$12r^2$$ is equal to _______

The point of intersection of the normals to the parabola $$y^2 = 4x$$ at the ends of its latus rectum is :

A hyperbola whose transverse axis is along the major axis of the conic $$\frac{x^2}{3} + \frac{y^2}{4} = 4$$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $$\frac{3}{2}$$, then which of the following points does not lie on the hyperbola?

The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $$\frac{x^2}{4} - \frac{y^2}{5} = 1$$, meets the x-axis and y-axis at A and B, respectively. Then $$OA^2 - OB^2$$, where O is the origin, equals:

The length of the chord of the parabola $$x^2 = 4y$$ having equation $$x - \sqrt{2}y + 4\sqrt{2} = 0$$ is: