If a circle drawn by assuming a chord parallel to the transverse axis of hyperbola x 2 a 2 - y 2 b 2 = 1 as diameter always passes through 2,0 , then

Question

If a circle drawn by assuming a chord parallel to the transverse axis of hyperbola x 2 a 2 - y 2 b 2 = 1 as diameter always passes through 2,0 , then, a = b = 2, b ≠ a, b = a = 1, b = a = 3

MARKS App · Accepted Answer

Let, the end points of the diameter of circle are asec ⁡ θ , b tan ⁡ θ & - asec ⁡ θ , b tan ⁡ θ ⇒ Equation of circle is x - a sec ⁡ θ x + a sec ⁡ θ + y - b tan ⁡ θ 2 = 0 ⇒ x 2 - a 2 sec 2 ⁡ θ + y 2 + b 2 tan 2 ⁡ θ - 2 b tan ⁡ θ y = 0 Because 2,0 satisfies above equation, hence, 4 - a 2 1 + tan 2 ⁡ θ + 0 + b 2 tan 2 ⁡ θ = 0 ⇒ 4 - a 2 + b 2 - a 2 tan 2 ⁡ θ = 0 Which is always true if a = 2 & b = 2

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