Question

Let the points $P$ and $Q$ lie on the lines $y=x$ and $y=2x$ respectively. The equation of the locus of the mid-point of the line segment joining $P$ and $Q,$ if $\left|P Q\right|=4,$ is

• A. $25x^{2}+36xy+13y^{2}=4$
• B. $25x^{2}-36xy+13y^{2}=4$
• C. $25x^{2}-36xy-13y^{2}=4$
• D. $25x^{2}+36xy-13y^{2}=4$

Answer 1

Answer: (B)

Let, $P\left(a , a\right)$ and $Q\left(b , 2 b\right)$ and the midpoint of $P$ $Q$ is $\left(h , k\right)$
$2h=a+b$
$2k=a+2b$
$a=2\left(2 h - k\right)$ and $b=2\left(k - h\right)$
Now, $\left(a - b\right)^{2}+\left(a - 2 b\right)^{2}=16$
$\Rightarrow \left(6 h - 4 k\right)^{2}+\left(8 h - 6 k\right)^{2}=16$
$\Rightarrow 9h^{2}+4k^{2}-12hk+16h^{2}+9k^{2}-24hk=4$
$\Rightarrow 25x^{2}+13y^{2}-36xy=4$