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Let A,B and C be three distinct points on `y^(2) = 8x` such that normals at these points are concurrent at P. The slope of AB is 2 and abscissa of centroid of `Delta ABC` is `(4)/(3)`. Which of the following is (are) correct? (a) Area of `DeltaABC` is 8 sq. units (b) Coordinates of `P -= (6,0)` (c) Angle between normals are `45^(@),45^(@),90^(@)` (d) Angle between normals are `30^(@),30^(@),60^(@)`

A

Area of `DeltaABC` is 8 sq. units

B

Coordinates of `P -= (6,0)`

C

Angle between normals are `45^(@),45^(@),90^(@)`

D

Angle between normals are `30^(@),30^(@),60^(@)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
Let `A = (2t_(1)^(2),4t_(1)), B = (2t_(2)^(2), 4t_(2))` and `C = (2t_(3)^(2),4t_(3))`
Slope of `AB = 2 rArr t_(1) + t_(2) =1` and `t_(1) + t_(2) + t_(3) =0` So, `t_(3) =-1`
Also, `(2(t_(1)^(2)+t_(2)^(2)+t_(3)^(2)))/(3) =(4)/(3) rArr t_(1)^(2)+t_(2)^(2) =1`
`rArr t_(1) =1, t_(2) =0`
`A = (2,4), B =(0,0)` and `C = (2,-4)`
Hence `P = (6,0)`
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