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Three points A,B and C taken on rectangu...

Three points A,B and C taken on rectangular hyperbola `xy = 4` where `B(-2,-2)` and `C(6,2//3)`. The normal at A is parallel to BC, then

A

circumcentre of `DeltaABC` is `(2,-2//3)`

B

equation of circumcircle of `DeltaABC` is `3x^(2)+3y^(2) -12x + 4y - 40 =0`

C

orthocenter of `DeltaABC` is `((2)/(sqrt(3)),2sqrt(3))`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
Given `B(-2,-2)` and `C(6,2//3)` on hyperbola
Let `A(2t,2//t)`
Differentiating curve `xy' +y =0` or `y' =- y//x =- 1//t^(2)`
Slope of normal at A is `t^(2)`
Normal is parallel to BC, which has slope `((2)/(3)+2)/(6+2)=(1)/(3)=t^(2)`
`:. t= (1)/(sqrt(3))`
`:. A ((2)/(sqrt(3)),2sqrt(3))`
Also (slope of AB) `xx` (Slope of `AC) =-1`
`:. DeltaABC` is a right angled `Delta`.
Circum-center of `DeltaABC` is mid-point `BC,(2,-2//3)`
Circum-circle of `DeltaABC` is circle on BC as diameter. Orthocenter of `DeltaABC` is at A.
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