Let C be the circle of radius unity centred at the origin. If two positive numbers `x_1 and x_2` are such that the line passing through `(x_1,-1) and (x_2, 1)` is tangent to C then `x_1*x_2`

Options

A. `x_(1)x_(2) = 1`
B. `x_(1)x_(2) =- 1`
C. `x_(1) +x_(2) = 1`
D. `4x_(1)x_(2)=1`

Answer

A

Explanation

Equation of circle is `x^(2)+y^(2) =1`
The equation of tangent to circle at any point `(cos theta, sin theta)` is `x cos theta +y sin theta =1`.
Since it is passing through points `(x_(1),-1)` and `(x_(2),1)` we have
`x_(1) cos theta - sin theta =1` or `x_(1) cos theta =1 +sin theta` (i)
and `x_(2) cos theta =1 - sin theta` (ii)
Multiplying the results, we have
`x_(1)x_(2) cos^(2) theta =1 -sin^(2) theta`
`rArr x_(1)x_(2) cos^(2) theta =cos^(2) theta`
`rArr x_(1)x_(2) = 1`