If the tangents and normals at the extremities of a focal chord of a parabola intersect at `(x_1,y_1)` and `(x_2,y_2),` respectively, then (a)`x_1=y_2` (b) `x_1=y_1` (c)`y_1=y_2` (d) `x_2=y_1`

Options

A. `x_(1)=x_(2)`
B. `x_(1)=y_(2)`
C. `y_(1)=y_(2)`
D. `x_(2)=y_(1)`

Answer

C

Explanation

Let `P(at_(1)^(2),2at_(1))" and Q"(at_(2)^(2), at_(2))` be a focal chord of the parabola `y^(2)=4ax`. The tangents at P and Q intersect at `(at_(1), t_(2),a(t_(1)+t_(2)))`
`:." "x_(1)=at_(1)t_(2)" and "y_(1)=a(t_(1)+t_(2))`
`rArr" "x_(1)=-a" and "y_(1)=a(t_(1)+t_(2))" "[{:(because" PQ is ","focal chord"),( ,:.t_(1)t_(2)=-1):}]`
Tghe normals at P and Q intersect at
`(2a+a(t_(1)^(2)+t_(2)^(2)+t_(1)t_(2)),-at_(1)t_(1)(t_(1)+t_(2))`
`:." "x_(2)=2a+a(t_(1)^(2)+t_(2)^(2)+t_(1)t_(2))" and "y_(2)=-at_(2)t_(2)(t_(1)+t_(2))`
`rArr" "x_(2)=a+a(t_(1)^(2)+t_(2)^(2))" and, "y_(2)=a(t_(1)+t_(2))`
Clearly, `y_(1)=y_(2)`.