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JEE MainMathematicsApplication of Derivatives

The normal to the curve y = x^2 2 at a point P in the first quadrant passes through the point Q(4, 1) . This normal intersects the curve again at a point R . If A is the area of the region bounded by the curve and the line segment PR , then the value of 12A is equal to _____.

Correct answer

125

Step-by-step solution

Let the point P on the curve be (h, h^2 2 ) with h > 0 . The derivative of the curve is dy dx = x . The slope of the normal at P is m = - 1 h . The equation of the normal at P is: y - h^2 2 = - 1 h (x - h) Since the normal passes through Q(4, 1) , we substitute x = 4 and y = 1 : 1 - h^2 2 = - 1 h (4 - h) 1 - h^2 2 = - 4 h + 1 h^2 2 = 4 h h^3 = 8 h = 2 So, P is (2, 2) . The equation of the normal line is: y - 2 = - 1 2 (x - 2) x + 2y = 6 y = 6-x 2 To find the intersection point R , we solve the line and curve equati

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