If p and q are the lengths of the perpendicul | vedclass

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

Question

(A) The first line is $x \operatorname{cosec} \alpha - y \sec \alpha = k \cot 2 \alpha$,which can be written as $\frac{x}{\sin \alpha} - \frac{y}{\cos

Vedclass · Accepted Answer

$4 p^{2} + q^{2}$

Vedclass · Answer

(A) The first line is $x \operatorname{cosec} \alpha - y \sec \alpha = k \cot 2 \alpha$,which can be written as $\frac{x}{\sin \alpha} - \frac{y}{\cos \alpha} = \frac{k \cos 2 \alpha}{\sin 2 \alpha}$.Multiplying by $\sin \alpha \cos \alpha$,we get $x \cos \alpha - y \sin \alpha = \frac{k \cos 2 \alpha}{\sin 2 \alpha} \cdot \sin \alpha \cos \alpha = \frac{k \cos 2 \alpha}{2 \sin \alpha \cos \alpha} \cdot \sin \alpha \cos \alpha = \frac{k}{2} \cos 2 \alpha$.The perpendicular distance $p$ from the origin $(0, 0)$ to the line $x \cos \alpha - y \sin \alpha - \frac{k}{2} \cos 2 \alpha = 0$ is $p = \left| \frac{-\frac{k}{2} \cos 2 \alpha}{\sqrt{\cos^{2} \alpha + \sin^{2} \alpha}} \right| = \left| \frac{k}{2} \cos 2 \alpha \right|$.Thus,$2p = |k \cos 2 \alpha|$,which implies $4p^{2} = k^{2} \cos^{2} 2 \alpha$ $(i)$.The second line is $x \sin \alpha + y \cos \alpha = k \sin 2 \alpha$. The perpendicular distance $q$ from the origin is $q = \left| \frac{-k \sin 2 \alpha}{\sqrt{\sin^{2} \alpha + \cos^{2} \alpha}} \right| = |k \sin 2 \alpha|$.Thus,$q^{2} = k^{2} \sin^{2} 2 \alpha$ $(ii)$.Adding $(i)$ and $(ii)$,we get $4p^{2} + q^{2} = k^{2} (\cos^{2} 2 \alpha + \sin^{2} 2 \alpha) = k^{2}$.

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines,$x \operatorname{cosec} \alpha - y \sec \alpha = k \cot 2 \alpha$ and $x \sin \alpha + y \cos \alpha = k \sin 2 \alpha$ respectively,then $k^{2}$ is equal to :

Similar Questions

$A$ line passes through the origin and is perpendicular to two given lines $2x + y + 6 = 0$ and $4x + 2y - 9 = 0$. What is the ratio in which the origin divides the segment formed by the intersection points of this line with the two given lines?

The number of integral points (integral point means both the coordinates should be integers) exactly in the interior of the triangle with vertices $(0, 0)$,$(0, 21)$,and $(21, 0)$ is:

The base $BC$ of a triangle $ABC$ is bisected at the point $(p, q)$ and the equations to the sides $AB$ and $AC$ are respectively $px + qy = 1$ and $qx + py = 1$. Then the equation to the median through $A$ is

$A$ line is such that its segment between the straight lines $5x - y - 4 = 0$ and $3x + 4y - 4 = 0$ is bisected at the point $(1, 5)$,then its equation is

The portion of the line $4x + 5y = 20$ in the first quadrant is trisected by the lines $L_1$ and $L_2$ passing through the origin. The tangent of an angle between the lines $L_1$ and $L_2$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module