If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?
JEE · Math · previous-year question
- A.(1, 1, 3)correct
- B.(1, 1, 0)
- C.$$\left( {{1 \over 2},2,0} \right)$$
- D.$$\left( {{1 \over 2},2,3} \right)$$
Answer
A. (1, 1, 3)
Explanation
Normal to the two given curves are y = m(x – c) – 2bm – bm3, y = mx – 4am – 2am3 If they have a common normal, then (c + 2b)m + bm3 = 4am + 2am3 $$ \Rightarrow $$ (4a – c – 2b) m = (b – 2a)m3 $$ \Rightarrow $$ (4a – c – 2b) = (b – 2a)m2 $$ \Rightarrow $$ m2 = $${c \over {2a - b}} - 2$$ $$>$$ 0 By checking all options we found (A) is right option. Note : We get that all the options are correct for m = 0 i.e., when common normal is x-axis. But may be in question they want common normal other than x – axis, hence answer is (A).
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