(B) The condition for the line $y = mx + c$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $c^2 = a^2m^2 - b^2$.Given the

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(B) The condition for the line $y = mx + c$ to be a tangent to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is $c^2 = a^2m^2 - b^2$.Given the line $y = \alpha x + \beta$,we have $m = \alpha$ and $c = \beta$.Substituting these into the condition,we get $\beta^2 = a^2\alpha^2 - b^2$.Replacing $(\alpha, \beta)$ with $(x, y)$,the locus is $y^2 = a^2x^2 - b^2$,which can be rewritten as $a^2x^2 - y^2 = b^2$.This equation represents a hyperbola.

Class 11 Mathematics 10-2. Parabola, Ellipse, Hyperbola Hyperbola

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

AIEEE 2005 All AIEEE

A
$A$ parabola
B
$A$ hyperbola
C
An ellipse
D
$A$ circle

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10-2. Parabola, Ellipse, Hyperbola

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Hyperbola

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If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $C$ are such that $CP$ is perpendicular to $CQ$,where $a < b$,then the value of $\frac{1}{(CP)^2} + \frac{1}{(CQ)^2}$ is:

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

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Find the equation of the hyperbola satisfying the given conditions: Vertices $(\pm 2, 0)$,foci $(\pm 3, 0)$.

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=b^2$ at four points $(x_1, y_1)$,$(x_2, y_2)$,$(x_3, y_3)$,and $(x_4, y_4)$,then $y_1 y_2 y_3 y_4 = $

The number of possible tangents which can be drawn to the curve $4x^2 - 9y^2 = 36$,which are perpendicular to the straight line $5x + 2y - 10 = 0$ is

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If two points $P$ and $Q$ on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with centre $C$ are such that $CP$ is perpendicular to $CQ$,where $a < b$,then the value of $\frac{1}{(CP)^2} + \frac{1}{(CQ)^2}$ is:

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