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9.Straight Line
hard
Locus of the image of point $ (2,3)$ in the line $\left( {2x - 3y + 4} \right) + k\left( {x - 2y + 3} \right) = 0,k \in R$ is a:
A
circle of radius $\sqrt 3 $
B
straight line parallel to $x- $ axis
C
straight line parallel to $y- $ axis
D
circle of radius $\;\sqrt 2 $
(JEE MAIN-2015)
Solution
Let $M$ be mid-point of $B B^{\prime}$ and $A M$ is $\perp$ bisector of $B B^{\prime}$
(where $A$ is the point of intersection of the given lines)
$(x-2)(x-1)+(y-2)(y-3)=0$
$\Rightarrow\left(\frac{h+2}{2}-2\right)\left(\frac{h+2}{2}-1\right)+\left(\frac{k+3}{2}-2\right)\left(\frac{k+3}{2}-3\right)=0$
$\Rightarrow(h-2)(h)+(k-1)(k-3)=0$
$\Rightarrow \quad x^{2}-2 x+y^{2}-4 y+3=0$
$\Rightarrow \quad(x-1)^{2}+(y-2)^{2}=2$
Standard 11
Mathematics
