From points on the straight line 3x-4y + 12 = 0, tangents are drawn to the circle `x^2 +y^2 = 4`. Then, the chords of contact pass through a fixed point. The slope of the chord of the circle having this fixed point as its mid-point is

Options

A. `(4)/(3)`
B. `(1)/(2)`
C. `(1)/(3)`
D. none of these

Answer

D

Explanation

Any point on the given line can be taken as `(alpha, (3alpha+12)/(4))`
Then the equation of the chord of contact of tangents from this point to `x^(2) +y^(2) = 4` is
`x alpha +y ((3alpha +12))/(4) - 4 =0`
or `alpha (4x +3y) +(12 y -16) =0`
For different values of `alpha`, the above chord passes through `P(-1,(4)/(3))` which is the point of intersection of `4x +3y = 0` and `3y -4 =0`.
Now slope of line joining center of circle `O(0,0)` and point `P(-1,(4)/(3))` is `-4//3`.
Thus, slope of the chord for which P is mid-point is `3//4`.