The locus of the image of the point (2, 3) in | vedclass

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

Question

(D) Let $P = (2, 3)$ be the given point and $P' = (h, k)$ be its image in the line $L_k: (2x - 3y + 4) + k(x - 2y + 3) = 0$.The line $L_k$ passes thro

Vedclass · Accepted Answer

circle of radius $\sqrt{2}$

Vedclass · Answer

(D) Let $P = (2, 3)$ be the given point and $P' = (h, k)$ be its image in the line $L_k: (2x - 3y + 4) + k(x - 2y + 3) = 0$.The line $L_k$ passes through the intersection of $L_1: 2x - 3y + 4 = 0$ and $L_2: x - 2y + 3 = 0$.Solving $L_1$ and $L_2$,we get $x = 1, y = 2$. Let this point be $A = (1, 2)$.Since $P'$ is the image of $P$ in $L_k$,the line $AP$ is perpendicular to $AP'$,and $AP = AP'$.The slope of $AP$ is $m_{AP} = \frac{3-2}{2-1} = 1$.Let $P' = (x, y)$. The slope of $AP'$ is $m_{AP'} = \frac{y-2}{x-1}$.Since $AP \perp AP'$,the product of their slopes is $-1$,so $\frac{y-2}{x-1} = -1 \implies y - 2 = -(x - 1) \implies x + y = 3$.Also,the distance $AP = \sqrt{(2-1)^2 + (3-2)^2} = \sqrt{2}$.Since $AP = AP'$,we have $(x-1)^2 + (y-2)^2 = AP^2 = 2$.Thus,the locus is a circle with center $(1, 2)$ and radius $\sqrt{2}$.

The locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0, k \in R$ is a:

Similar Questions

What is the equation of the locus of a point which moves such that $4$ times its distance from the $x$-axis is the square of its distance from the origin?

Let a point $P$ be such that its distance from the point $(5, 0)$ is thrice the distance of $P$ from the point $(-5, 0)$. If the locus of the point $P$ is a circle of radius $r$,then $4r^{2}$ is equal to ...... .

The locus of the centre of a circle which cuts the circles $x^2 + y^2 + 4x - 6y + 9 = 0$ and $x^2 + y^2 - 4x + 6y + 4 = 0$ orthogonally is

The line segment joining the points $(-5, 1)$ and $(3, 2)$ subtends a right angle at a variable point $P$. Find the locus of the point $P$.

Two perpendicular tangents to the circle $x^2 + y^2 = a^2$ meet at point $P$. The equation of the locus of $P$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module