Question

Let \( E \) be an ellipse whose major axis is the X-axis and minor axis is the Y-axis. If the distance of a point \( \left(\frac{5}{2}, 2\sqrt{3}\right) \) on \( E \) from its foci are \( \frac{7}{2} \) and \( \frac{13}{2} \), then the eccentricity of the ellipse \( E \) is:

• A. \( \frac{3}{5} \)
• B. \( \frac{1}{5} \)
• C. \( \frac{1}{\sqrt{5}} \)
• D. \( \frac{1}{\sqrt{2}} \)

Hint:

In ellipses, the sum of distances from any point on the ellipse to the two foci equals the length of the major axis. Use this to determine \( a \), and then apply the relation \( c^2 = a^2 - b^2 \).

Answer 1

The Correct Option is A

In an ellipse, the sum of distances from any point on the ellipse to the two foci equals \( 2a \) (major axis length). \[ \frac{7}{2} + \frac{13}{2} = 10 \Rightarrow 2a = 10 \Rightarrow a = 5 \] Let the coordinates of the foci be \( (\pm c, 0) \). Given the point \( P = \left(\frac{5}{2}, 2\sqrt{3}\right) \), apply the distance formula and solve for \( c \). Eventually, using the relation \( c^2 = a^2 - b^2 \), and \( e = \frac{c}{a} \), we get: \[ e = \frac{3}{5} \]