Maths · Sections of conics, equations of conic sections (parabola, ellipse and hyperbola) in standard forms
if the distance between the foci is equal to the length of the latus-rectum. Fin
if the distance between the foci is equal to the length of the latus-rectum. Find the eccentricity of the ellipse.
- A. \( \frac{\sqrt{5}-1}{2} \)
- B. \( \frac{\sqrt{5}+1}{2} \)
- C. \( \frac{\sqrt{5}-1}{4} \)
- D. None of these
Step-by-step solution
For an ellipse, distance between foci = 2ae, length of latus rectum = 2b^2/a. Given equality: 2ae = 2b^2/a ⇒ ae = b^2/a ⇒ b^2 = a^2 e. Using b^2 = a^2(1 - e^2), we get a^2(1 - e^2) = a^2 e ⇒ 1 - e^2 = e ⇒ e^2 + e - 1 = 0. Solving gives e = (√5 - 1)/2 (positive root less than 1).
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