Question

3. If the parabolas \( y^{2}=4 b(x-c) \) and \( y^{2}=8 a x \) have a common normal, then which one of the following is a valid choice for the ordered triad \( ( a , b , c ) \) ? (2019) 1) \( (1,1,3) \) 2) \( (1,1,0) \) 3) \( \left(\frac{1}{2}, 2,3\right) \) 4) \( \left(\frac{1}{2}, 2,0\right) \)

Answer 1

The correct option is (1) (1, 1, 3).

If m is the slope of the common normal, then the equation of normal to parabola y² = 4b(x − c) is given by

y = m(x − c) − 2bm − bm³ ⋯(1)

Equation of normal to parabola y² = 8ax is

y = mx − 4am − 2am³ ⋯(2)

From equations (1) and (2), we get

m(x − c) − 2bm−bm³ = mx − 4am − 2am³

⇒ m²(2a − b) = c − 2(2a−b)

Hence (1,1,3) satisfies equation.