Definition of a circle

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The pole of a straight line with respect to the circle $x^{2} + y^{2} = a^{2}$ lies on the circle $x^{2} + y^{2} = 9 a^{2}$ If the straight line touches the circle $x^{2} + y^{2} = r^{2}$ then
$\frac{a^{2}}{r^{2}}$ is equal to

Moderate

Solution

Let $(α, β)$ be the pole of the given straight line with respect to the circle $x^{2} + y^{2} = a^{2}$ Then, the equation of the polar is
$α x + β y - a^{2} = 0$
It is given that $(α, β)$ lies on the circle $x^{2} + y^{2} = a^{2}$
$∴ α^{2} + β^{2} = 9 a^{2}$
Since line in (i) touches the circle $x^{2} + y^{2} = r^{2}$
$∴ |\frac{- a^{2}}{\sqrt{α^{2} + β^{2}}}| = r \Rightarrow \frac{a^{2}}{\sqrt{9 a^{2}}} = r \Rightarrow 9 r^{2} = a^{2}$

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Definition of a circle

Remember concepts with our Masterclasses.

The pole of a straight line with respect to the circle x2+y2=a2 lies on the circle x2+y2=9a2 If the straight line touches the circle x2+y2=r2 thena2r2 is equal to

Let (α,β) be the pole of the given straight line with respect to the circle x2+y2=a2 Then, the equation of the polar isαx+βy−a2=0It is given that (α,β) lies on the circle x2+y2=a2∴ α2+β2=9a2Since line in (i) touches the circle x2+y2=r2∴−a2α2+β2=r⇒a29a2=r⇒9r2=a2

Ready to Test Your Skills?

free study material

The pole of a straight line with respect to the circle $x^{2} + y^{2} = a^{2}$ lies on the circle $x^{2} + y^{2} = 9 a^{2}$ If the straight line touches the circle $x^{2} + y^{2} = r^{2}$ then
$\frac{a^{2}}{r^{2}}$ is equal to