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Question:

An ellipse intersects the hyperbola 2x² - y² = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then what is the equation of the ellipse?

the foci of ellipse are(±√2,o)

equation of ellipse isx2+2y2=2

the foci of ellipse are(±1,0)

equation of ellipse isx2+2y2=4

Solution:

Ellipse and hyperbola will be confocal ⇒ (±ae,0) = (±1,0) ⇒ (±a × 1/√2,0) = (±1,0) ⇒ a=√2 and e=1/√2 ⇒ b² = a²(1-e²) ⇒ b² = 1 ∴Equation of ellipse is x²/2 + y²/1 = 1