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Properties of Determinants

If the polar ...

Question

If the polar of $(p, q)$ with respect to the circle $x^{2} + y^{2} = a^{2}$ touches the circle ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ then ${(h p + k q - a^{2})}^{2} =$

A

r^{2} (p^{2} + q^{2})

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B

r (p^{2} + q^{2})

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C

r^{2} (p^{2} - q^{2})

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D

none of these

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Solution

The correct option is C $r^{2} (p^{2} + q^{2})$
If the polar of $(p, q)$ with respect to the circle $x^{2} + y^{2} = a^{2}$ is $p x + q y - a^{2} = 0$
Since, polar touches the cirlce ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$
then radius = perpendicular distance from center to polar
$r = ∣ ∣ ∣ ∣ \frac{p h + q k - a^{2}}{\sqrt{p^{2} + q^{2}}} ∣ ∣ ∣ ∣$
$\Rightarrow {(p h + q k - a^{2})}^{2} = r^{2} (p^{2} + q^{2})$
Ans: A

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