Hyperbola practice · Medium

The straight line $x + y = \sqrt 2 p$ will touch the hyperbola $4{x^2} - 9{y^2} = 36,$ if

The locus of the middle points of the portions of the tangents of the hyperbola. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ included between the axes is

The locus of middle points of chords of hyperbola $3{x^2} - 2{y^2} + 4x - 6y = 0$ parallel to $y = 2x$ is

If $5{x^2} + \lambda {y^2} = 20$ represents a rectangular hyperbola, then $\lambda$ equals

Consider the set of hyperbola $xy = k,k \in R.$ Let ${e_1}$ be the eccentricity when $k = 4$ and ${e_2}$ be the eccentricity when $k = 9$, then ${e_1} - {e_2}$ is equal to

The shortest distance between the line $y=x$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ is

If the chords of contacts of the tangents from the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ to the hyperbola $2 x^2-3 y^2=6$ are at right angle, then $4 x_1 x_2+9 y_1 y_2$ is equal to

$P$ is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1,\;N$ is the foot of the perpendicular from $P$ on the transverse axis. The tangent to the hyperbola at $P$ meets the transverse axis at $T$. If $O$ is the centre of the hyperbola, then $OT.ON$ is equal to

If $t$ is a parameter, then $x = a\left( {t + \frac{1}{t}} \right),y = b\left( {t - \frac{1}{t}} \right)$ represents

The equation of tangent to $4 x^2-9 y^2=36$ which are perpendicular to straight line $5 x+2 y-10=0$ are

For some $\theta \in\left(0, \frac{\pi}{2}\right)$ if eccentricity of the hyperbola $x^2-y^2 \operatorname{cosec}^2 \theta=8$ is $\sqrt{7}$ times the eccentricity of the ellipse $x^2+y^2 \operatorname{cosec}^2 \theta=4$, then the length of the latus rectum of the ellipse is

The asymptotes of the hyperbola $xy = hx + ky$ are

The equation $16 x^2-3 y^2-32 x+12 y-44=0$ represents a hyperbola

The equation of a hyperbola, conjugate to the hyperbola $x^2+3 x y+2 y^2+2 x+3 y=0$ is

If the chord of contact of tangents from a point $P$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends a right angle at the centre, then the locus of ${P}$ is

Find the equation of the hyperbola whose directrix is $2 x+y=1$, focus is $(1,1)$ and eccentricity is $\sqrt{3}$.

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

The locus of a point $P\left( {\mathrm{\alpha },\mathrm{\beta }} \right)$ moving under the condition that the line $y = \mathrm{\alpha }x + \mathrm{\beta }$ is a tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, is

The condition that a straight line with slope $m$ will be normal to parabola $y^2=4 a x$ as well as a tangent to rectangular hyperbola $x^2-y^2=a^2$ is

Tangent to hyperbola $x y=c^2$ at point $P$ intersects the $x$-axis at $T$ and the $y$-axis at $T^{\prime}$. Normal to hyperbola at ${P}$ intersects the ${x}$-axis at ${N}$ and the ${y}$-axis at ${N}^{\prime}$. If the area of the triangles $PNT$ and $PN^{\prime}T^{\prime}$ are $\Delta$ and $\Delta^{\prime}$ respectively then $\frac{1}{\Delta}+\frac{1}{\Delta^{\prime}}$, is equal to

If the circle ${x^2} + {y^2} = {a^2}$ intersects the hyperbola $xy = {c^2}$ in four points $P\left( {{x_1},{y_1}} \right),Q\left( {{x_2},{y_2}} \right),R\left( {{x_3},{y_3}} \right)$ and $S\left( {{x_4},{y_4}} \right)$, then

A common tangent to $x^2-2 y^2=18$ and $x^2+y^2=9$ is

If tangent to the hyperbola $\frac{x^2}{4}-y^2=1$ cut the circle $x^2+y^2=16$ at two distinct point $A$ and $B$, then the locus of mid-point of $A$ and $B$ is

The tangent to the hyperbola $x y=c^2$ at point $P(t)$ intersects the $X$-axis at $T$ and the $Y$-axis. at $T^{\prime}$. The normal to the hyperbola at $P(t)$ intersects the $X$-axis at $N$ and the $Y$-axis at $N^{\prime}$. The areas of the triangles $P N T$ and $P N^{\prime} T^{\prime}$ are $\Delta$ and $\Delta^{\prime}$ respectively, then $\frac{1}{\Delta}+\frac{1}{\Delta^{\prime}}$ is

Equation of the chord of the hyperbola $25 x^2-16 y^2=400$ which is bisected at the point $(6,2)$ is

If the chords of contact of tangents drawn from $P$ to the hyperbola ${x^2} - {y^2} = {a^2}$ and its auxiliary circle are at right angle, then $P$ lies on

The product of perpendicular distances from any point on the hyperbola $9{x^2} - 16{y^2} = 144$ to its asymptotes is

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

A hyperbola has the asymptotes $x + 2y = 3$ and $x - y = 0$ and passes through $\left( {2,1} \right)$. Its centre is

Normal at point $(5,3)$ to the rectangular hyperbola $x y-y-2 x-2=0$ meets the curve at the point whose coordinates are