Question

If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then the eccentricity of the conic is

${\text{A}}{\text{. }}\dfrac{{\sqrt {386} }}{{12}} \\\ {\text{B}}{\text{. }}\dfrac{{\sqrt {386} }}{{13}} \\\ {\text{C}}{\text{. }}\dfrac{{\sqrt {386} }}{{25}} \\\ {\text{D}}{\text{. }}\dfrac{{\sqrt {386} }}{{38}} \\\$

Answer 1

Hint:To find the answer, we find the distance between foci. Then we find the sum of distances from foci and then find the eccentricity by using relevant formulae.

Complete step-by-step answer:
And let P (0, 0) be a point on the conic.

SP = $\sqrt {{{\left( {5 - 0} \right)}^2} + {{\left( {12 - 0} \right)}^2}} = \sqrt {25 + 144}$ = 13

S′P = $\sqrt {{{\left( {24 - 0} \right)}^2} + {{\left( {7 - 0} \right)}^2}} = \sqrt {576 + 49}$ = 25

SS′ = $\sqrt {{{\left( {24 - 5} \right)}^2} + {{\left( {7 - 12} \right)}^2}} = \sqrt {192 + 52}$=$\sqrt {386}$

If the conic is an ellipse, then

SP + S′P = 2a and SS′ = 2ae, where a is the foci and e is the eccentricity

e = $\dfrac{{{\text{SS'}}}}{{{\text{SP + S'P}}}}$
= $\dfrac{{\sqrt {386} }}{{13 + 25}}$
= $\dfrac{{\sqrt {386} }}{{38}}$
Hence Option D is the correct answer.

Note:The key in solving such types of problems is finding the distance between foci and the sum of distances from foci. And knowing the formulae in ellipse respectively is a crucial step. Distance between two points (x, y) and (a, b) is given by D =$\sqrt {{{\left( {{\text{x - a}}} \right)}^2} + {{\left( {{\text{y - b}}} \right)}^2}}$.