A point P (\sqrt{2} sin A, cot B) lies on the curve (x + y)2 + 4(y – x)2 = 8. If minimum value of \left(\left(\right. 2 tanC - \sqrt{2} sinA \left.\right)\right)^{2} + \left(\left(\right. 2 cotC - cotB \left.\right)\right)^{2} = a + b \sqrt{2} \textrm{ } , \textrm{ }\textrm{ } when \textrm{ }\textrm{ } C \in R - \left{\right. \frac{nπ}{2} \left.\right} , n , a , b \in I, then find the value of (a + b).

Engineering

Mathematics

Rectangular Hyperbola

Question

A point P ( $\sqrt{2}$ sin A, cot B) lies on the curve (x + y)² + 4(y – x)² = 8. If minimum value of
${(2 tanC - \sqrt{2} sinA)}^{2} + {(2 cotC - cotB)}^{2} = a + b \sqrt{2}, when C \in R - {\frac{nπ}{2}}, n, a, b \in I$ , then find the value of
(a + b).

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Clearly P lies on given ellipse and

$∵$ (2tan C, 2cot C) lies on rectangular hyperbola xy = 4

$∴ Minimum value = {(2 \sqrt{2} - 2)}^{2}$

$= 12 - 8 \sqrt{2} = a + b \sqrt{2}$

$∴$ a + b = 4.