KVPY Sample Paper KVPY Stream-SX Model Paper-7

If the parabolas \[{{y}^{2}}=4b(x-c)\] and \[{{y}^{2}}=8ax\] have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?

A) \[\left( \frac{1}{2},2,3 \right)\]

B) (1, 1, 3)

C) \[\left( \frac{1}{2},2,0 \right)\]

D) (1, 1, 0)

Correct Answer: B

Solution :

Normal to there 2 curves are

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

\[y=mx-4am-2a{{m}^{3}}\]

If they how a common normal

\[(c+2b)-4a=(2a-b){{m}^{2}}\]

(\[m=0\]corresponds to axis)

\[\Rightarrow \] \[{{m}^{2}}=\frac{c}{2a-b}-2>0\]

\[\frac{c}{2a-b}>2\]

According to the question answer is (1, 2, 3, 4) but may be in question they want common normal other than x-axis, hence answer is [B].