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The tangents to the curve y = (x - 2)^(2...

The tangents to the curve `y = (x - 2)^(2) - 1` at its points of intersectio with the line x - y = 3, intersect at the point

A

`((5)/(2), 1)`

B

`(-(5)/(2), -1)`

C

`((5)/(2), -1)`

D

`(-(5)/(2), 1)`

Text Solution

Verified by Experts

The correct Answer is:
C
Given equation of parabola is
`y = (x - 2)^(2) - 1`
`implies y = x^(2) - 4x + 3`
Now, let `(x_(1) y_(1))` be the print of intersection of tangents of parabola (i) and line x - y = 3, then
Equation of chord of contact of point `(x_(1), y_(1))` w.r.t
parabola (i) is
`T = 0`
`implies (1)/(2) (y + y_(1) = xx_(1) - 2 (x + x_(1)) + 3`
`implies y + y_(2) = 2x (x_(1) - 2) - 4x_(1) + 6`
`implies 2x (x_(1) - 2) - y = 4 x_(1) + y_(1) - 6`, this equation represent the line x - y = 3 only, so on comparing, we get
`*(2(x_(1) - 2))/(1) = (-1)/(-1) = (4x_(1) + y_(1) - 6) `
`implies x_(1) = (5)/(2)` and `y_(1) = 1`
So, the required point is `((5)/(2), - 1)`
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