If the chord xcos⁡α$$+ysin$$⁡α−$$p=0$$ of the circle $$x2+y2$$−$$a2=0$$ subtends a right angle at the centre of the circle, $$thena2p2$$ is equal to

Question

If the chord xcos⁡α$$+ysin$$⁡α−$$p=0$$ of the circle $$x2+y2$$−$$a2=0$$ subtends a right angle at the centre of the circle, $$thena2p2$$ is equal to

Infinity Learn · Accepted Answer

The combined equation of the lines joining the origin to the points of intersection of $x \cos α + y \sin α = p$ and $x^{2} + y^{2} - a^{2} = 0$ is a homogeneous equation of second degree given by

$\begin{array}{l} x^{2} + y^{2} - a^{2} {(\frac{x \cos α + y \sin α}{p})}^{2} = 0 \\ \Rightarrow x^{2} (p^{2} - a^{2} \cos^{2} α) + y^{2} (p^{2} - a^{2} \sin^{2} α) - (a^{2} \sin 2 α) x y = 0 \end{array}$

The lines given by this equation are at right angle.

$Coeff. of x^{2} + Coeff. of y^{2} = 0 \Rightarrow p^{2} - a^{2} \cos^{2} α + p^{2} - a^{2} \sin^{2} α = 0 \Rightarrow 2 p^{2} = a^{2}$

Definition of a circle

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If the chord $x \cos α + y \sin α - p = 0$ of the circle $x^{2} + y^{2} - a^{2} = 0$ subtends a right angle at the centre of the circle, then $\frac{a^{2}}{p^{2}}$ is equal to

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Definition of a circle

Remember concepts with our Masterclasses.

If the chord xcos⁡α+ysin⁡α−p=0 of the circle x2+y2−a2=0 subtends a right angle at the centre of the circle, thena2p2 is equal to

Ready to Test Your Skills?

free study material

If the chord $x \cos α + y \sin α - p = 0$ of the circle $x^{2} + y^{2} - a^{2} = 0$ subtends a right angle at the centre of the circle, then $\frac{a^{2}}{p^{2}}$ is equal to