The line $p x + q y = r$ touches the hyperbola $b^{2} x^{2} - a^{2} y^{2} = a^{2} b^{2}$ if:

a^{2} p^{2} + b^{2} q^{2} = r^{2}

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a^{2} p^{2} - b^{2} q^{2} = r^{2}

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a^{2} q^{2} + b^{2} p^{2} = r^{2}

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a^{2} q^{2} - b^{2} p^{2} = r^{2}

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Solution

The correct option is D $a^{2} p^{2} - b^{2} q^{2} = r^{2}$
Given equations may be written as,
$y = - \frac{p}{q} x - \frac{r}{q}$ and $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
Now using formula for tangency of a line to a hyperbola $c^{2} = a^{2} m^{2} - b^{2}$
$\Rightarrow \frac{r^{2}}{q^{2}} = a^{2} \frac{p^{2}}{q^{2}} - b^{2}$
$\Rightarrow a^{2} p^{2} - b^{2} q^{2} = r^{2}$
Hence, option 'B' is correct.

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Similar questions

If $p x + q y + r = 0$ is tangent of the parabola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ then

p x + q y = 1

x^{2} + y^{2} = r^{2}

(p, q)

x^{2} + y^{2} = r^{- 2}

△ A B C

a^{2} b^{2} {sin}^{2} C = a^{2} p^{2} + b^{2} q^{2} - 2 a b p q cos C

p x + q y

x y = r^{2}

b^{2} x^{2} - a^{2} y^{2} = a^{2} b^{2}

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