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A(9,6), B(4,-4) lie on the parabola such...

A(9,6), B(4,-4) lie on the parabola such that area of `DeltaABC` is maximum (given "C" lies at the arc contain origin). Find the maximum area of `Delta ABC` (A) `101/2` (B) `69/2` (C) `125/4` (D) `521/3`

A

`31(1)/(4)`

B

32

C

`31(3)/(4)`

D

`30(1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A
Accroding to given iformation, we have the following figure,

For `y^(2)=4ax`, parametric coordinates of a point is `(at^(2) 2at)`.
`:.` For `y^(2)=4x`, let coordinates of C be `(t^(2),2t)`
Then are of `DeltaABC=(1)/(2)||{:(t^(2),2t,1),(9,6,1),(4,-4,1):}||`
`=(1)/(2)|t^(2)(6-(-4))-2t(9-4)+1(-36-24)|`
`=(1)/(2)|10t^(2)-10t-60|=(10)/(2)|t^(2)-t-6|=5|t^(2)-t-6|`
Let, `A(t)=5|t^(2)-t-6|" "...(i)`
Clearly , `A(4,-4)-=A(t_(1)^(2),2t_(1))) rArr 2t_(2)=-'4`
`rArr t_(1)=-2`
and `B(9,6)-=B(t_(2)^(2),2t_(2)) rAr 2t_(2)=6 rarr t_(2)=3`
Sicne,S is on the are AOB, the parameter 't' for point `C in (-2,3)`
Let `f(t)=t^(2)-t-6rArr f'(t)=0rArr t=(1)/(2)`
Thus, for A(t), critical points is at `t=(1)/(2)`
Now, `A((1)/(2))=5|((1)/(2))^(2)-(1)/(2)-6|=(125)/(4)=31(1)/(4)` [ Using Eq. (i)]
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