First slide

Ellipse

Question

$The eccentricity of the locus of point (3 h + 2, k), where (h, k) lies on the circle x^{2} + y^{2} = 1, is$

Moderate

A

$\frac{1}{3}$
B

$\frac{\sqrt{2}}{3}$
C

$\frac{2 \sqrt{2}}{3}$
D

$\frac{1}{\sqrt{3}}$

Solution

$\begin{array}{l} Let p = 3 h + 2 and q = k \\ or h = \frac{p - 2}{3} and k = q \end{array}$
$Since (h, k) lies on x^{2} + y^{2} = 1, we have h^{2} + k^{2} = 1$
$or {(\frac{p - 2}{3})}^{2} + q^{2} = 1$
$The locus is {(\frac{x - 2}{3})}^{2} + y^{2} = 1$
$which has eccentricity e = \sqrt{1 - \frac{1}{9}} = \frac{2 \sqrt{2}}{3}$

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