Question

If a, b, c are distinct positive real numbers such that the parabolas y² = 4ax and y² = 4c (x– b) will have a common normal, then

(a)   0 < b/a - c < 1

(b)   b/a - c < 0

(c)  1 <b/a - c < 2

(d)  b/a - c > 2

Answer 1

Correct option  (d)  b/a - c > 2

Explanation:

Equation of normals are

y = mx –2am – am³ ......(1)

y = m(x – b) – 2cm – cm³ ......(2)

Equation 1 and 2 are identical then

–2am – am³ = –bm –2cm –cm³ / m

2a + am² = b + 2c + cm²

(a– c)m² = b + 2(c  – a)

m² = b/a - c - 2

m = ± √b/a - c - 2

For m be real b/a - c  > 2