If the ellipse $x^{2} + k^{2}y^{2} = k^{2}a^{2}$ is confocal with the hyperbola $x^{2} - y^{2} = a^{

Question

If the ellipse $x^{2} + k^{2}y^{2} = k^{2}a^{2}$ is confocal with the hyperbola $x^{2} - y^{2} = a^{2}(k > 1)$. Then which of the following statement(s) is/are correct?

Accepted Answer

Sol. Given ellipse: x² + k²y² = k²a²⇒ x²/(k²a²) + y²/a² = 1Thus, for ellipse: A = ka, B = a, C = a√(k² − 1)Given hyperbola: x² − y² = a²Its focal distance is: C₁ = a√2Since the ellipse and hyperbola are confocal,a√(k² − 1) = a√2 ⇒ k² = 3 ⇒ k = √3Now check the options:(a) eₑ/eₕ = [√2/√3]/√2 = 1/√3 ✓(b) Major axis / transverse axis = (2√3a)/(2a) = √3 ✓(c) Minor axis / conjugate axis = 2a/2a = 1 ≠ √3 ✗(d) Latus rectum ratio = [2a²/(√3a)]/[2a] = 1/√3 ✓

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