Question

The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^2}{4}+\frac{y^2}{9}=1$ is :

• A. $9x^2+4y^2+18x+8y+145=0$
• B. $36x^2+16y^2+90x+56y+145=0$
• C. $36x^2+16y^2+108x+80y+145=0$
• D. $36x^2+16y^2+72x+32y+145=0$

Answer 1

Answer: (C)

General point on $\frac{x^2}{4}+\frac{y^2}{9}=1$ is $A(2\cos \theta ,3\sin \theta )$ given $B(-3,-5)$
midpoint $C\left(\frac{2\cos \theta -3}{2},\frac{3\sin \theta -5}{2}\right)$
$h=\frac{2\cos \theta -3}{2};k=\frac{3\sin \theta -5}{2}$
$\Rightarrow {\left(\frac{2h+3}{2}\right)}^2+{\left(\frac{2k+5}{3}\right)}^2=1$
$\Rightarrow 36x^2+16y^2+108x+80y+145=0$