Conic Sections Questions (164)

The range of $a$ for which the points $(a, 2 + a)$ and $(\frac{3}{2}, a, a^2)$ lie on opposite sides of the line $2x + 3y = 6$ can lie in intervals:

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:

The normal drawn at the extremities $P$ and $Q$ of a focal chord meet the parabola again in $P'$ and $Q'$ respectively. Then:

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 $O$ be the vertex of a parabola and $Q$ be any point on the axis of the parabola. If $PQR$ be any chord passing through $Q$ and $PM$ and $RN$ be the ordinates of $P$ and $R$, then:

Coordinates of the feet of normal drawn from the point $(7, 14)$ to the parabola $x^2 - 8x - 16y = 0$ is/are:

The values of $a$ for which $y = ax^2 + ax + \frac{1}{24}$, $x = ay^2 + ay + \frac{1}{24}$ touch each other is/are

If $x, y \in \mathbb{R}$ then the equation $3x^2 - 2(9y + 8)x^2 + (361y^2 + 2(100 + y^3)x + 64) = 2(190y + 2y^2)$ represents in rectangular Cartesian system:

For the hyperbola $9y^2 - 16y^2 - 18x + 32y - 151 = 0$

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

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

If the focus of the parabola $x^2 - ky + 3 = 0$ is $(0, 2)$, then a value of $k$ is/are:

If P is a point on a hyperbola, then

The values of $a$ for which $y = ax^2 + ax + \frac{1}{24}$, $x = ay^2 + ay + \frac{1}{24}$ touch each other is/are

Let $PQ$ be a chord of the parabola $y^2 = 4x$. A circle drawn with $PQ$ as a diameter passes through the vertex $V$ of the parabola. If area $(APQ) = 20 \text{ unit}^2$ then the coordinates of $P$ is/are

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

Variable circle is described to pass through point $(1, 0)$ and tangent to the curve $y = \tan(\tan^{-1} x)$. The locus of the centre of the circle is a parabola whose:

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

From the points $(x_1, y_1)$ and $(x_2, y_2)$ tangents are drawn to the hyperbola $xy = c^2$, such that a circle passes through these points and the four points of contact, then:

Equation of asymptotes are

Variable pairs of chords at right angles are drawn through any point $P$ (with eccentric angle $\frac{\pi}{4}$) on the ellipse $\frac{x^2}{4} + y^2 = 1$, to meet the ellipse at two points say $A$ and $B$. If the line joining $A$ and $B$ passes through a fixed point $Q(a,b)$ such that $a^2 + b^2$ has the value equal to $\frac{m}{n}$, where $m, n$ are respectively prime positive integers, then the value of $\frac{m+n}{3}$ is____.

If the normals to curve $y = x^2$ at the points $P, Q$ and $R$ pass through the point $\left(0, \frac{3}{2}\right)$, then the radius of the circle circumscribing $\triangle PQR$ is ______.

If the equation on reflection of $\frac{(x-4)^2}{16} + \frac{(y-3)^2}{9} = 1$ about the line $x - y - 2 = 0$ is $16x^2 + 9y^2 + k_1x - 36y + k_2 = 0$ then $\frac{k_1 + k_2}{100}$ is ______.

If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, $S_1$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1S_2S_3S_4$ is:

Area of the triangle formed by the asymptotes of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and any tangent to the hyperbola is $a^2\tan\lambda$ in magnitude then its eccentricity is:

The chord $AC$ of the parabola $y^2 = 4ax$ subtends an angle of $90°$ at points $B$ and $D$ on the parabola. If $A, B, C$ and $D$ are represented by $t_1, t_2, t_3$ & $t_4$, then find the value of $\left|\frac{t_2 + t_4}{t_1 + t_3}\right|$.

The length of the major axis of the ellipse $(5x - 10)^2 + (5y + 15)^2 = \frac{(3x - 4y + 7)^2}{4}$ is $A$. Find $[A]$. ($[|$ represents greatest integer function$)$

Find the number of distinct normals that can be drawn to the ellipse $\frac{x^2}{69} + \frac{y^2}{25} = 1$ from the point $P(0, 6)$.

If $k$ be the length of the latus rectum of the hyperbola $16x^2 - 9y^2 + 32x + 36y - 164 = 0$, then find $3k/8$.

If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is 4 and conjugate axis length is 6, $S_1$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1S_2S_3S_4$ is:

Match the following: P. The normal at an end of a latusrectum of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passes through an end of the minor axis if $e^4$ is equal to Q. PQ is a double ordinate of a parabola $y^2 = 4ax$. If the locus of its point of trisection is another parabola length of whose latus rectum is k times the length of the latus rectum of the given parabola then k is equal to R. If e and $e'$ are the distances of the extremities of any focal chord from the focus f of the parabola $y^2 = 4ax$, then $\frac{1}{e} + \frac{1}{e'}$ is equal to S. If e and $e'$ be the eccentricities of a hyperbola and its conjugate, then $\frac{1}{e} + \frac{1}{e'^2}$ is equal to

The centre of the conic $C$ is:

From a point $P$, perpendicular tangents are drawn to the ellipse $x^2 + 2y^2 = 2$. If the chords of contact are tangents to a family of concentric circles, having the centres same as that of the ellipse then the ratio of the areas of the largest circle to the smallest circle is____.

If the ellipse $\frac{x^2}{4} + y^2 = 1$ meets the ellipse $x^2 + \frac{y^2}{a^2} = 1$ in four distinct points and $a = b^2 - 5b + 7$, then $b$ does not lie in

Let $S$ be the focus of $y^2 = 4x$ and a point $P$ is moving on the curve such that its abscissa is increasing at the rate of 4 units/sec, then the rate of increase of projection of $SP$ on $x + y = 1$ when $P$ is at $(4, 4)$ is:

From any point $P$ lying in first quadrant on the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, $PN$ is drawn perpendicular to the major axis and produced to $Q$ so that $NQ$ equals $PS$, where $S$ is the focus $(-3, 0)$. Then the locus of $Q$ is:

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