The condition that the line x cos alpha + y s | vedclass

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(C) For a line to touch a circle,the perpendicular distance from the center of the circle to the line must be equal to the radius of the circle.The eq

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${p^2} = {a^2}$

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(C) For a line to touch a circle,the perpendicular distance from the center of the circle to the line must be equal to the radius of the circle.The equation of the circle is ${x^2} + {y^2} = {a^2}$,so its center is $(0, 0)$ and its radius is $a$.The perpendicular distance from the center $(0, 0)$ to the line $x \cos \alpha + y \sin \alpha - p = 0$ is given by:$d = \frac{|(0)\cos \alpha + (0)\sin \alpha - p|}{\sqrt{\cos^2 \alpha + \sin^2 \alpha}}$Since $\cos^2 \alpha + \sin^2 \alpha = 1$,we have:$d = \frac{|-p|}{\sqrt{1}} = |p|$Setting the distance equal to the radius $(d = a)$:$|p| = a$Squaring both sides,we get:${p^2} = {a^2}$

The condition that the line $x \cos \alpha + y \sin \alpha = p$ may touch the circle ${x^2} + {y^2} = {a^2}$ is

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