The locus of the midpoint of the line segment joining the point (4,3) and points on the el…

Question

The locus of the midpoint of the line segment joining the point $(4,3)$ and points on the ellipse $x^2+2y^2=4$ is an ellipse with eccentricity:

Accepted Answer

Point on ellipse: $(2\cos\theta,\sqrt2\sin\theta)$. Midpoint with $(4,3)$: $X=2+\cos\theta,\ Y=\dfrac{3+\sqrt2\sin\theta}2$.

$$(X-2)^2+\frac{(2Y-3)^2}2=1\ \Longrightarrow\ \frac{(X-2)^2}1+\frac{(Y-1.5)^2}{1/2}=1.$$

$$e=\sqrt{1-\frac{1/2}1}=\frac1{\sqrt2}.$$

Solutions · 1