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

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 $C\left(\frac{2 \cos \theta-3}{2}, \frac{3 \sin \theta-5}{2}\right)$
$h =\frac{2 \cos \theta-3}{2} ; k =\frac{3 \sin \theta-5}{2}$
$\Rightarrow \left(\frac{2 h +3}{2}\right)^{2}+\left(\frac{2 k +5}{3}\right)^{2}=1$
$\Rightarrow 36 x^{2}+16 y^{2}+108 x+80 y+145=0$