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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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 +

Vedclass · Accepted Answer

$a^2$

Vedclass · Answer

(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 $4P_1^2 + P_2^2$?

Similar Questions

If the perpendicular distance from the point $(1, 1)$ to the line $3x + 4y + c = 0$ is $7$,then the possible values of $c$ are:

The distance between the lines given by $3x + 4y = 9$ and $6x + 8y = 15$ is (in $units$)

The equations of the lines passing through the point $(1, 0)$ and at a distance $\frac{\sqrt{3}}{2}$ from the origin are:

If the line $2x - y + 3 = 0$ is at a distance of $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4x - 2y + \alpha = 0$ and $6x - 3y + \beta = 0$ respectively,then the sum of all possible values of $\alpha$ and $\beta$ is:

Let the lines be $L_1: 2x + 3y - 7 = 0$ and $L_2: 2x + 3y - 12 = 0$. For the point $A(3, -5)$,which of the following is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module