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Let a circle C have its centre at the vertex of the parabola y^2 = 16x and pass through its focus. A line L with slope m is tangent to the given parabola and intersects the circle C to form a chord of length 4 2 . Then the value of 15m^2 + 10 is equal to

Correct answer

25

Step-by-step solution

For the parabola y^2 = 16x , a = 4 . The vertex is (0,0) and the focus is (4,0) . The circle C has its centre at (0,0) and passes through (4,0) , so its radius is r = 4 . The equation of the circle C is x^2 + y^2 = 16 . The equation of the tangent to the parabola with slope m is y = mx + 4 m , which can be written as m^2x - my + 4 = 0 . The perpendicular distance d from the centre (0,0) to the tangent is d = 4 m^4 + m^2 . The length of the chord is given by 2 r^2 - d^2 = 4 2 . 16 - d^2 = 2 2 16 - d^2 = 8 d^2 = 8 .

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