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The length of focal chord of the parabola y^2=4ax at a distance b from the vertex is c then

The length of focal chord of the parabola y^2=4ax at a distance b from the vertex is c then

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Grade:11

2 Answers

Deepak Kumar Shringi
askIITians Faculty 4404 Points
6 years ago
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Rupam Bende
24 Points
2 years ago
:-   Find intersections A(x1, y1), B(x2, y2) of P & L
                        Eliminate y from (1) & (2):
                        m2x2 - 2am x + m2 a2 = 4ax 
                        m2 x2-2a(m + 2) x + m2 a2 = 0                     ..............(3)
                        x1, x2 are the roots, x1 + x2 = 2a (m+2)/m2 ;; x1 x2 = a2         .........(4)
                        Eliminate x from (1) & (2):
                        y = m(y2/4a) - ma
                        my2 - 4a y-4a2 m = 0                        ..........(5)
                        y1, y2 are the roots, y1 + y2 = 4a/m     ;; y1 y2 = - 4a2            ........(6)
                        Now length of focal chord = c = AB
                        c2 = (x1 - x2)2 + (y1 -y2)2
                        = (x1 + x2)2 - 4x1 x2 + (y1 + y2)2 - 4y1 y2
                        =4a2 (m4 + 4 + 4m2)/m4 - 4a2 + 16a2/m2+16a2
                        = 16a2 (m2 +1)2/m4
                        = 16 a2 (a2 / b2)2            .............using (3)
                            b2c = 4a3

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