question_answer

The locus of a point which divides the line segment joining the point \[(0,-1)\] and a point on the parabola, \[{{x}^{2}}=4y,\] internally in the ratio \[1:2,\] is     [JEE MAIN Held On 08-01-2020 Morning]

A) \[9{{x}^{2}}-12y=8\]

B) \[4{{x}^{2}}-3y=2\]

C) \[{{x}^{2}}-3y=2\]

D) \[9{{x}^{2}}-3y=2\]

Correct Answer: A

Solution :

Let \[P\equiv (2t,{{t}^{2}})\] Given

By section formula \[\frac{(0)(2)+2t(1)}{3}=h\]        (i) and

\[\frac{(-1)(2)+{{t}^{2}}(1)}{3}=k\]           (ii) by (i) and (ii)

\[\Rightarrow 3k+2={{\left( \frac{3h}{2} \right)}^{2}}\]

\[\Rightarrow 12y+8=9{{x}^{2}}\]