Consider a circle x - α 2 + y - β 2 = 50 , where α , β 0 . If the circle touches the line y + x = 0 at the point P , whose distance from the origin is 4 2 , then ( α + β ) 2 is equal to _______.

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Consider a circle x - α 2 + y - β 2 = 50 , where α , β > 0 . If the circle touches the line y + x = 0 at the point P , whose distance from the origin is 4 2 , then ( α + β ) 2 is equal to _______.

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Given: x - α 2 + y - β 2 = 50 The given equation of circle represent centre as α , β and radius as 5 2 units. Now, x + y = 0 is tangent to the given circle at P . We know that, radius is perpendicular to tangent at the point of tangency. ⇒ C P ⊥ x + y = 0 and C P = r ⇒ r = α × 1 + β × 1 1 2 + 1 2 ⇒ 5 2 = α + β 2 ⇒ 50 = α + β 2 2 ⇒ α + β 2 = 100

Consider a circle x - α 2 + y - β 2 = 50 , where α , β > 0 . If the circle touches the line y + x = 0 at the point P , whose distance from the origin is 4 2 , then ( α + β ) 2 is equal to _______.

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