Conic Sections Questions (164)

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

For which of the following hyperbolas, we can have more than one pair of perpendicular tangents?

From point (2, 2) tangents are drawn to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ then point of contact lies in

For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, let $n$ be the number of points on the plane through which perpendicular tangents are drawn.

Equations of the asymptotes of the hyperbola whose equation is given by $x = a \tan(\theta + \alpha)$ and $y = b \tan(\theta + \beta)$, $\theta$ being a parameter, is/are:

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:

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

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 locus of a point on the variable parabola $y^2 = 4ax$, whose distance from focus is constant $k$, is equal to: ($a$ is parameter)

An equilateral triangle $SAB$ is inscribed in the parabola $y^2 = 4ax$ having its focus at '$S$'. If chord $AB$ lies towards the left of $S$, then side length of this triangle is:

$\min \left[(x_1 - x_2)^2 + \left|12 - \sqrt{1 - x_1^2} - \sqrt{4x_2}\right|^2\right], \forall x_1, x_2 \in \mathbb{R}$ is:

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:

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

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:

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.

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:

Tangents are drawn to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ and the circle $x^2 + y^2 = a^2$ at the points where a common ordinate cuts them (on the same side of the $x$-axis). Then the greatest acute angle between these tangents is given by:

If the eccentricity of the hyperbola $x^2 - y^2\sec^2 a = 5$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2\sec^2 a + y^2 = 25$, then a value of $a$ is:

If a ray of light incident along the line $3x + (5 - 4\sqrt{2})y = 15$ gets reflected from the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $(4\sqrt{2}, 3)$, then its reflected ray goes along the line:

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

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:

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:

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

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:

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

If origin is shifted to point $\left(3, \frac{7}{2}\right)$ and the axes are rotated through an angle $\theta$ in clockwise sense so that equation of given hyperbola changes to the standard form $\frac{x'^2}{a^2} - \frac{y'^2}{b^2} = 1$, then $\theta$ is:

Let $A\left(\frac{1}{2}, 0\right), B\left(\frac{3}{2}, 0\right), C\left(\frac{5}{2}, 0\right)$ be the given points and $P$ be a point satisfying $\max(PA + PB, PB + PC) < 2$. All points $P$ are points common to:

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

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

The extremities of latus rectum of a parabola are $(1, 1)$ and $(1, -1)$, then the equation of the parabola can be :

Parabola $y^2 = 4x$ and the circle having its centre at $(6, 5)$ intersect at right angle. Possible point of intersection of these curves can be :

A normal drawn to parabola $y^2 = 4ax$ meet the curve again at $Q$ such that angle subtended by $PQ$ at vertex is $90°$, then coordinates of $P$ can be :

If a pair of variable straight lines $x^2 + 4y^2 + \alpha xy = 0$ (where $\alpha$ is a real parameter) cut the ellipse $x^2 + 4y^2 = 4$ at two points A and B, then the locus of the point of intersection of tangents at A and B is:

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

Consider the ellipse $\frac{x^2}{t(k^2 + 2k + 5)} + \frac{y^2}{t(k+1)} = 1$ and $f(x)$ is a positive decreasing function, then:

If the tangent drawn at point $(t^2, 2t)$ on the parabola $y^2 = 4x$ is same as the normal drawn at point $(\sqrt{5}\cos\theta, 2\sin\theta)$ on the ellipse $4x^2 + 5y^2 = 20$. Then:

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

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 which of the following hyperbolas, we can have more than one pair of perpendicular tangents?

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

If P is a point on a hyperbola, then

From point (2, 2) tangents are drawn to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ then point of contact lies in

For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, let $n$ be the number of points on the plane through which perpendicular tangents are drawn.

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:

Equations of the asymptotes of the hyperbola whose equation is given by $x = a \tan(\theta + \alpha)$ and $y = b \tan(\theta + \beta)$, $\theta$ being a parameter, is/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:

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

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:

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: