Conic Sections Questions (164)

Let $P, Q$ be two points on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ whose eccentric angles differ by a right angle. The tangents at $P$ and $Q$ meet at $R$. If the chord $PQ$ divides the line segment $CR$ in $m:n$, then find $m/n$ (where $C$ is the centre of ellipse).

'$O'$ is the vertex of parabola $y^2 = 4x$ and $L$ is the upper end of latus rectum. If $LH$ is drawn perpendicular to $OL$ meeting $x$-axis in $H$, then length of double ordinate through $H$ is $\sqrt{N}$, then $N =$

If the normals to the curve $y = x^2$ at the points $P, Q$ & $R$ passes through the point $(0, 3/2)$, find the radius of the circle circumscribing $\triangle PQR$.

$$\frac{x^2}{r^2 - r - 6} + \frac{y^2}{r^2 - 6r + 5} = 1$$ will represents the ellipse, if $r$ lies in the interval

Consider a parabola $4y = x^2$ and point $B(0,1)$. Let $A_1\left(x_1, y_1\right), A_2\left(x_2, y_2\right), \ldots\ldots\ldots\ldots, A_n\left(x_n, y_n\right)$ are $n$ points on the parabola such that $x_r > 0$ and $\angle OBA_r = \frac{r\pi}{2n}$ $(r = 1, 2, \ldots\ldots\ldots, n)$ then $\pi\left(\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{n} BA_r\right)$ is equal to ______.

If the normal at the points where the straight line $lx + my = 1$ meet the parabola $y^2 = 4ux$, meet at the point $(h, k)$ on the parabola $y^2 = 4ux$, then $\frac{kl}{am}$ is equal to____.

A normal to the hyperbola $\frac{x^2}{4} - \frac{y^2}{1} = 1$ has equal intercepts on positive $x$ and $y$-axes. If this normal touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then find $[a^2 + b^2]$. ($[|$ represents greatest integer function$)$

There are exactly two points on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ whose distance from its centre is same and is equal to $\sqrt{\frac{a^2 + 2b^2}{2}}$. Then the eccentricity of the ellipse is:

If two distinct tangents can be drawn from the point $(a, 2)$ on different branches of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, then

The ratio of latus rectum of given parabola and that of made by locus of point $P$ is:

The straight line $\frac{x}{4} + \frac{y}{3} = 1$ intersects the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ at two points $A$ and $B$, there is a point $P$ on this ellipse such that the area of $\triangle PAB$ is equal to $6\left(\sqrt{2} - 1\right)$. Then the number of such points $P$ is ______.

A normal is drawn to the ellipse $\frac{x^2}{(a^2+2a+1)^2} + \frac{y^2}{(a^2+1)^2} = 1$, $a > 0$ whose centre is at $O$. If maximum radius of the circle, centered at the origin and touching the normal, is $5$ then the positive value of $'a'$ is:....

If the equation of parabola is $y^2 = 8x$, then locus of $P$ is:

Angle subtended by $AB$ at centre of the hyperbola is

If the point $\left(\lambda^2, \lambda - 2\right)$ is a point lying interior of the region bounded by the parabola $y^2 = 2x$ and the chord joining the point $(2, 2)$ and $(8, -4)$, then the number of the integral values of $\lambda$ is ______.

The tangents at a point $P$ to the rectangular hyperbola $xy = 1$ meets the lines $x - y = 0$ and $x + y = 0$ at $Q$ and $R$ respectively and $\Delta_1$ is the area of the triangle $OQR$, where $O$ is the origin. The normal at $P$ meets $X$-axis at $M$ and the $Y$-axis at $N$ and $\Delta_2$ is the area of the triangle $OMN$, then the value of $\Delta_1^2\Delta_2$ is \ldots.

The straight line $\frac{lx}{a} + \frac{my}{b} = n$ meet the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at in the points $P$ and $Q$. If $OP$ and $OQ$ are along a pair of semi-conjugate diameters, $O$ being the centre of the ellipse, then $\frac{l^2}{n^2} + \frac{m^2}{n^2}$ equals____.

Transverse and conjugate axes of a rectangular hyperbola are along $X$-axis and $Y$-axis respectively and the distance between the foci is $10\sqrt{14}$. Number of the points $(x, y)$ on the curve such that $x$ and $y$ are positive integers, is equal to ______.

Equation of the tangent to the hyperbola at $\left(-1, -\frac{1}{2}\right)$ is

The straight line $ax + by + c = 0$ cuts the locus of point of intersection of the lines $\frac{tx}{3} + t = 0, \frac{x}{4} + \frac{ty}{3} - t = 0$ at $A$ & $B$ such that line $AB$ subtends a right angle at the origin, then $\left[\frac{3a - 4b}{c}\right]$ is _____. ($[|$ represents greatest integer function$)$

At any point $P$ on the parabola $y^2 - 2y - 4x + 5 = 0$, a tangent is drawn which meets the directrix at $Q$. If the locus of $R$ which divides $QP$ externally in the ratio $1:2$ is $(y-1)^2(x+1)+\lambda = 0$, then find $\lambda$.

Two tangents to a parabola are $x - y = 0$ and $x + y = 0$. If $(2, 3)$ is focus of the parabola, then the equation of tangent at vertex is:

The tangent to the hyperbola $xy = 1$ at the point $P$ intersects the $X$-axis at $T$ and the $Y$-axis at $T'$. The normal to the hyperbola at $P$ intersects the $X$-axis at $N$ and the $Y$-axis at $N'$. Let the areas of the triangles $PNT$ and $PN'T'$ are $\Delta$ and $\Delta'$ respectively. If $\frac{1}{\Delta} + \frac{1}{\Delta'}$ is constant for all positions of $P$ then the value of constant is ______.

The locus of $P$ is symmetric about:

A chord cut the same branch of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ in $P, P'$ and the asymptotes in $Q, Q'$, then the value of $(PQ + PQ') - (P'Q' + P'Q)$ is____.

A normal to the hyperbola $\frac{x^2}{4} - \frac{y^2}{1} = 1$, has equal intercepts on positive $x$ and positive $y$ axis. If the normal touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, then the value of $\frac{9}{25}(a^2 + b^2)$ is____.

The length of the sub-tangent to the hyperbola $x^2 - 4y^2 = 4$ corresponding to the normal having slope unity is $\frac{1}{\sqrt{k}}$, then $k$ is equal to ______.

Locus of intersection of two perpendicular tangents to the hyperbola is:

Equation of asymptotes are

Tangent is drawn at any fixed point $(x_1, y_1)$ on the parabola $y^2 = 4ux$. Now tangents are drawn from any point on this tangent to the circle $x^2 + y^2 = a^2$ so that all the chords of contact pass through a fixed point $(x_2, y_2)$. If $4\left(\frac{x_1}{x_2}\right) + \left(\frac{y_1}{y_2}\right)^2 = ka^2$, then $k$ equals to

If $P = (x_1, y_1)$ the slope of third normal is:

A curve is represented by $C = 21x^2 - 6xy + 29y^2 + 6x - 58y - 151 = 0$. Eccentricity of curve is:

The chord $AB$ of the parabola $y^2 = 4ax$ cuts the axis of the parabola at $C$ (C is internal to AB). If $A = (at_1^2, 2at_1)$ and $B = (at_2^2, 2at_2)$ and $AC : AB = 1 : 3$, then:

Let $P(x, y)$ is a variable point such that $\sqrt{(x-1)^2 + (y-2)^2} - \sqrt{(x-5)^2 + (y-5)^2} = 3$ which represents hyperbola. The eccentricity $e'$ of the corresponding conjugate hyperbola is:

If the equation of parabola is $y^2 = 8x$, then locus of $P$ is:

The parabolas $y^2 = 4ax$ and $y^2 = 4c(x - d)$ have a common normal other than the X-axis if and only if:

Let $E_1$ and $E_2$ be two ellipses $\frac{x^2}{a^2} + y^2 = 1$ and $x^2 + \frac{y^2}{a^2} = 1$ (where $a$ is a parameter). Then the locus of the points of intersection of the ellipses $E_1$ and $E_2$ is a set of curves comprising

Let $y^2 = 4ax$ be a parabola and $PQ$ is a focal chord. Let $R$ be the point of intersection of the tangents at $P$ and $Q$, then:

Two tangents to a parabola are $x - y = 0$ and $x + y = 0$. If $(2, 3)$ is focus of the parabola, then the equation of tangent at vertex is:

If $y = 2$ be the directrix and $(0, 1)$ be the vertex of the parabola $x^2 + \lambda y + \mu = 0$ then :

The equation $(x - a)^2 + (y - b)^2 = k(lx + my + n)^2$ represents

The ratio of latus rectum of given parabola and that of made by locus of point $P$ is:

Circles are drawn on chords of the rectangular hyperbola $xy = 4$ parallel to the line $y = x$ as diameters. All such circles pass through two fixed points whose coordinates are

If equation of tangent at P, Q and vertex A of a parabola are $3x + 4y - 7 = 0$, $2x + 3y - 10 = 0$ and $x - y = 0$ respectively, then:

The area of region of the point $P$ is:

Let $S$ and $S'$ be two foci of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. If a circle described on $SS'$ as diameter intersects the ellipse in real and distinct points, then the eccentricity $e$ of the ellipse satisfies.

A curve is represented by $C = 21x^2 - 6xy + 29y^2 + 6x - 58y - 151 = 0$. Eccentricity of curve is:

The equation $\sqrt{x^2 + (y-1)^2} - \sqrt{x^2 + (y+1)^2} = K$ will represent a hyperbola for

The lengths of axes are:

The straight line joining any point $P$ on the parabola $y^2 = 4ax$ to the vertex and perpendicular from the focus to the tangent at $P$, intersect at $R$, then the equation of the locus of $R$ is: