question_answer

Let \[P(4,-4)\] and \[Q(9,6)\] be points on the parabola \[{{y}^{2}}=4a(x-b).\]. Let R be a point on the arc of the parabola between P and Q. Then the area of \[\Delta PQR\] is the largest when the point R is                                                            

A) \[(4,4)\]                 

B) \[(8,16)\]           

C) \[(1/4,1)\]      

D) None of these

Correct Answer: C

Solution :

[c] \[(4,-4)\] and \[(9,6)\] lie on \[{{y}^{2}}-4x(x-b)\] \[\Rightarrow \,\,\,\,\,16=4a(4-b)\] and \[36=4a(9-b)\] Solving these, we get \[a=1,\text{ }b=0\] So, equation of parabola is \[{{y}^{2}}=4x.\] Let the point R be  \[({{t}^{2}},2t)\], where \[t\in (-2,3)\]. \[=\frac{1}{2}\left| 10t-10{{t}^{2}}+60 \right|\,\,=\frac{1}{4}\,\,\left| 125-5{{(2t-1)}^{2}} \right|\] Thus, area is largest when \[t=\frac{1}{2}\] \[\therefore \,\,\,\,\,\,\,\,R({{t}^{2}},2t)=R(1/4,1)\]