JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Parabola

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)

(A) $(\frac{1}{2}, 2, 3)$
(B) $(1, 1, 3)$ ✓ Correct
(C) $(\frac{1}{2}, 2, 0)$
(D) $(1, 1, 0)$

Answer: (B)

Solution

A normal to

y^{2} = 8 a x

at slope

m

y = m x - 4 am - 2 a m^{3}

, and a normal to

y^{2} = 4 b (x - c)

y = m (x - c) - 2 bm - b m^{3}

. For these to be the same non-axial normal, equating the constant parts and cancelling

m

gives

m^{2} = \frac{c + 2 b - 4 a}{2 a - b}

, which must be positive for a real slope. Testing

(a, b, c) = (1, 1, 3)

2 a - b = 1

and

c + 2 b - 4 a = 1

, so

m^{2} = 1 > 0

, a genuine common normal, whereas the other triples force

m^{2} < 0

. So

(1, 1, 3)

is the valid choice.

JEE Main · 2019 · 10 Jan 2019 · Shift 1 · Single correct option

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