If the parabolas y2 = 4b (x – c) and y2 = 8ax have a common normal, then which one of the following is a valid choices for the ordered triad (a, b, c)?

Engineering

Mathematics

Normal to Parabola

Question

If the parabolas y² = 4b (x – c) and y² = 8ax have a common normal, then which one of the following is a valid choices for the ordered triad (a, b, c)?

$(\frac{1}{2}, 2, 0)$

(1, 1, 0)

(1, 1, 3)

$(\frac{1}{2}, 2, 3)$

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y = mx – 4am – 2am²

y = m (x – c) – 2bm – bm³

– 4am – 2am² = c + ab + bm²

(1) 4 + 2m² = 3 + 2 + m² m² = 1 possible

(2) 2 + m² = 0 + 4 + 2m² → not possible

(3) 2 + m² = 3 + 4 + 2m² → not possible

(4) 4 + 2m² = 0 + 2 + m² → not possible