Normals are drawn from the point P with slopes m 1 , m 2 , m 3 to the parabola y 2 = 4x. If locus of P with m 1 m 2 = α is a part of parabola itself, then find α.

Question

Normals are drawn from the point P with slopes m 1 , m 2 , m 3 to the parabola y 2 = 4x. If locus of P with m 1 m 2 = α is a part of parabola itself, then find α.

Arun · Answer

. 3 is y = mx – 2am – am = 4ax. 2 y The equation of the normal to the parabola Hence, the equation of the normal to the parabola y 2 = 4x is y = mx – 2m – m 3 . Now, if it passes through (h, k), then we have k = mh – 2m – m 3 This can be written as m 3 + m(2-h) + k = 0 …. (1) Here, m 1 + m 2 + m 3 = 0 m 1 m 2 + m 2 m 3 + m 3 m 1 = 2 - h Also, m 1 m 2 m 3 = -k, where m 1 m 2 = α Now, this gives m 3 = -k/α and this must satisfy equation (1). Hence, we have (-k/α) 3 + (-k/α)(2-h) + k = 0 Solving this, we get k 2 = α 2 h – 2α 2 + α 3 And so we have y 2 = α 2 x – 2α 2 + α 3 On comparing this equation with y 2 = 4x we get α 2 = 4 and -2α 2 + α 3 = 0 This gives α = 2.

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Normals are drawn from the point P with slopes m 1 , m 2 , m 3 to the parabola y 2 = 4x. If locus of P with m 1 m 2 = α is a part of parabola itself, then find α.

Normals are drawn from the point P with slopes m1, m2, m3 to the parabola y2 = 4x. If locus of P with m1m2 = α is a part of parabola itself, then find α.

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