Let equation of two diameters of a circle x ^{2}+ y ^{2} -2 x +2 fy +1=0 be 2 px

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10-2. Parabola, Ellipse, Hyperbola

normal

Let the equation of two diameters of a circle $x ^{2}+ y ^{2}$ $-2 x +2 fy +1=0$ be $2 px - y =1$ and $2 x + py =4 p$. Then the slope $m \in(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to $......$

$6$

$2$

$4$

$8$

(JEE MAIN-2022)

Solution

$2 p+f-1=0$

$2-p f-4 p=0$

$2=p(f+4)$

$p=\frac{2}{f+4}$

$2 p=1-f$

$\frac{4}{f+4}=1-f$

$f^{2}+3 f=0$

$f=0 \text { or }-3$

Hyperbola $3 x ^{2}- y ^{2}=3, x ^{2}-\frac{ y ^{2}}{3}=1$

$y=m x \pm \sqrt{m^{2}-3}$

It passes $(1,0)$

$o=m \pm \sqrt{m^{2}-3}$

$m$ tends $\infty$

$\text { It passes }(1,3)$

$3=m \pm \sqrt{m^{2}-3}$

$(3-m)^{2}=m^{2}-3$

$m=2$

Standard 11

Mathematics

Let $H : \frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1$, where $a > b >0$, be $a$ hyperbola in the $xy$-plane whose conjugate axis $LM$ subtends an angle of $60^{\circ}$ at one of its vertices $N$. Let the area of the triangle $LMN$ be $4 \sqrt{3}$..

List $I$ List $II$

$P$ The length of the conjugate axis of $H$ is $1$ $8$

$Q$ The eccentricity of $H$ is $2$ ${\frac{4}{\sqrt{3}}}$

$R$ The distance between the foci of $H$ is $3$ ${\frac{2}{\sqrt{3}}}$

$S$ The length of the latus rectum of $H$ is $4$ $4$

The correct option is:

normal

(IIT-2018)

Let $H_1: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 e _2^2$ is equal to _____

hard

(JEE MAIN-2025)

Find the equation of the hyperbola satisfying the give conditions: Foci $(0,\,\pm 13),$ the conjugate axis is of length $24.$

medium

Let $P$ is a point on hyperbola $x^2 -y^2 = 4$ , which is at minimum distance from $(0,-1)$ then distance of $P$ from $x-$ axis is

normal

The equation of a common tangent to the conics $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{a^2}}} – \frac{{{x^2}}}{{{b^2}}} = 1$ is

hard

List $I$	List $II$
$P$ The length of the conjugate axis of $H$ is	$1$ $8$
$Q$ The eccentricity of $H$ is	$2$ ${\frac{4}{\sqrt{3}}}$
$R$ The distance between the foci of $H$ is	$3$ ${\frac{2}{\sqrt{3}}}$
$S$ The length of the latus rectum of $H$ is	$4$ $4$

Let the equation of two diameters of a circle $x ^{2}+ y ^{2}$ $-2 x +2 fy +1=0$ be $2 px - y =1$ and $2 x + py =4 p$. Then the slope $m \in(0, \infty)$ of the tangent to the hyperbola $3 x^{2}-y^{2}=3$ passing through the centre of the circle is equal to $......$

Solution

Similar Questions

Find the equation of the hyperbola satisfying the give conditions: Foci $(0,\,\pm 13),$ the conjugate axis is of length $24.$

Let $P$ is a point on hyperbola $x^2 -y^2 = 4$ , which is at minimum distance from $(0,-1)$ then distance of $P$ from $x-$ axis is

The equation of a common tangent to the conics $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{a^2}}} – \frac{{{x^2}}}{{{b^2}}} = 1$ is

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