Question

S(3, 4) and S’(9, 12) are the foci of an ellipse and the foot of the perpendicular from S to a tangent to the ellipse is (1, –4). Then the eccentricity of the ellipse is

• A.
• B.
• C.
• D. none

Answer 1

The correct option is C

SS’ = 2ae $\Rightarrow$ 2ae = 10 $\Rightarrow$ ae = 5
Centre C is the midpoint of SS’. $\therefore$ C = (6, 8)
Equation of the auxiliary circle is $(x-6{)}^2+(y-8{)}^2=a^2$
Since the locus of the foot of the perpendicular from locus to a tangent of the ellipse is called the auxiliary circle, we have (1, –4) lies on the auxiliary circle.
$\therefore$ a = 13. Hence $e=\frac{5}{13}$