KVPY Sample Paper KVPY Stream-SX Model Paper-1

Let the curve C be the mirror image of the parabola \[{{y}^{2}}=4x\] with respect to the line \[x+y+4=0.\] If A and B are the points of intersection of C with the line \[y=-\,5,\] then the distance between A and B is

A) 8

B) 4

C) 5

D) 6

Correct Answer: B

Solution :

[b]

Let point \[P\,({{t}^{2}},2t)\] lie on parabola \[{{y}^{2}}=4x.\]Image of \[P\,({{t}^{2}},2t)\] with respect to line \[x+y+4=0\]is \[8\,(h,k).\]

\[\therefore \]\[\frac{h-{{t}^{2}}}{1}=\frac{k-2t}{1}=\frac{-\,2\,({{t}^{2}}+2t+4)}{2}\]

Hence, curve C becomes

\[\frac{{{(x+4)}^{2}}}{4}+4=-\,y\]

Since it intersect with \[y=-\,5\]

\[\therefore \] \[{{(x+4)}^{2}}+16=20\]

\[\Rightarrow \]\[{{(x+4)}^{2}}=4\]

\[\Rightarrow \]\[x=-\,2\]or \[x=6\]

\[\therefore \]\[A=\,(-2,-\,5)\]and \[B=\,(-\,6,-\,5)\]

\[\Rightarrow \]\[AB=\sqrt{{{(-2+6)}^{2}}+{{(-\,5+5)}^{2}}}=4\]