If the line $ \mathrm{y}+\mathrm{x}=0 $ bisects two chords drawn from a point $ \left(\frac{1+a \sqrt{2}}{2}, \frac{1-a \sqrt{2}}{2}\right) $ to the circle $ 2 x^{2}+2 y^{2}-(1+a \sqrt{2}) x-(1-a \sqrt{2}) y=0 $, then $ a $ lies in the interval $ (-\infty,-\lambda) \cup(\lambda, \infty) $, the numerical quantity $ \lambda $ should be equal to

Question

If the line $ \mathrm{y}+\mathrm{x}=0 $ bisects two chords drawn from a point $ \left(\frac{1+a \sqrt{2}}{2}, \frac{1-a \sqrt{2}}{2}\right) $ to the circle $ 2 x^{2}+2 y^{2}-(1+a \sqrt{2}) x-(1-a \sqrt{2}) y=0 $, then $ a $ lies in the interval $ (-\infty,-\lambda) \cup(\lambda, \infty) $, the numerical quantity $ \lambda $ should be equal to,

MARKS App · Accepted Answer

Equation of chord AB whose mid point is h , - h is T = S 1 2 xh - 2 yh - 1 + 2 a x + h 2 - 1 - 2 a y - h 2 = 2 h 2 + 2 h 2 - 1 + 2 a h + 1 - 2 a h ⇒ 4 xh - 4 yh - 1 + 2 a x + h - 1 - 2 a y - h = 8 h 2 - 2 1 + 2 a h + 2 1 - 2 a h ⇒ x 4 h - 1 + 2 a - y 4 h + 1 - 2 a - h 1 + 2 a + h 1 - 2 a = 8 h 2 - 2 1 + 2 a h + 2 1 - 2 a h or 8 h 2 - 1 + 2 a h + 1 - 2 a h - x 4 h - 1 + 2 a + y 4 h + 1 - 2 a = 0 It passes through A 1 + 2 a 2 , 1 - 2 a 2 then 8 h 2 - 2 2 ah - 1 + 2 a 2 4 h - 1 + 2 a + 1 - 2 a 2 4 h + 1 - 2 a or 8 h 2 - 2 2 ah - 2 h 1 + 2 a + 1 + 2 a 2 2 + 2 h 1 - 2 a + 1 - 2 a 2 2 = 0 or 8 h 2 - 6 2 ah + 1 + 2 a 2 = 0 Hence for two real and different values of h, we must have - 6 2 a 2 - 4 · 8 · 1 + 2 a 2 > 0 or a 2 - 4 > 0 a + 2 a - 2 > 0 ∴ A ∈ - ∞ , - 2 ∪ 2 , ∞

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