The angle between the two tangents from the origin to the circle `(x-7)^2+ (y+1)^2= 25` equals

Options

A. `(pi)/(4)`
B. `(pi)/(3)`
C. `(pi)/(2)`
D. `(2pi)/(3)`

Answer

C

Explanation

The equation of any line through the origin (0, 0) is y=mx.
If it is a tangent to the circle `(x-7)^(2)+(y+1)^(2)=5^(2)`, then
`|(7m+1)/(sqrt(m^(2)+1))|=5`
`rArr (7m+1)^(2)=25(m^(2)+1)rArr24m^(2)+14m-24=0 " " ...(i)`
This equation, being a quadratic in m, gives two values of m, say `m_(1)` and `m_(2)`. These two values of m are the slopes of the tangents drawn from he origin to the given circle.
From (i), we have `m_(1)m_(2)=-1`.
Hence, the two tangents are perpendicular.
`ul("ALITER")` Clearly, (0, 0) lies is the directior circle `(x-7)^(2)+(y+1)^(2)=50` of the given circle.
Hence, required angle is `pi//2`.