If P_1 and P_2 are the lengths of the perpend | vedclass

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(A) The length of the perpendicular from $(0, 0)$ to the line $x \sec 	heta + y \csc 	heta = a$ is given by:$P_1 = \frac{|-a|}{\sqrt{\sec^2 	heta +

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$a^2$

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(A) The length of the perpendicular from $(0, 0)$ to the line $x \sec 	heta + y \csc 	heta = a$ is given by:$P_1 = \frac{|-a|}{\sqrt{\sec^2 	heta + \csc^2 	heta}} = \frac{a}{\sqrt{\frac{1}{\cos^2 	heta} + \frac{1}{\sin^2 	heta}}} = \frac{a}{\sqrt{\frac{\sin^2 	heta + \cos^2 	heta}{\sin^2 	heta \cos^2 	heta}}} = a \sin 	heta \cos 	heta = \frac{a}{2} \sin 2	heta$.Thus,$2P_1 = a \sin 2	heta$.The length of the perpendicular from $(0, 0)$ to the line $x \cos 	heta - y \sin 	heta = a \cos 2	heta$ is given by:$P_2 = \frac{|-a \cos 2	heta|}{\sqrt{\cos^2 	heta + \sin^2 	heta}} = a \cos 2	heta$.Now,calculating $4P_1^2 + P_2^2$:$4P_1^2 + P_2^2 = (2P_1)^2 + P_2^2 = (a \sin 2	heta)^2 + (a \cos 2	heta)^2 = a^2 (\sin^2 2	heta + \cos^2 2	heta) = a^2$.

If $P_1$ and $P_2$ are the lengths of the perpendiculars from the origin to the lines $x \sec \theta + y \csc \theta = a$ and $x \cos \theta - y \sin \theta = a \cos 2\theta$ respectively,then what is the value of $4P_1^2 + P_2^2$?

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