Through the vertex `O` of the parabola `y^2=4a x` , two chords `O Pa n dO Q` are drawn and the circles on OP and OQ as diameters intersect at `Rdot` If `theta_1,theta_2` , and `varphi` are the angles made with the axis by the tangents at `P` and `Q` on the parabola and by `O R ,` then value of `cottheta_1+cottheta_2` is (a) `-2tanvarphi` (b) `-2tan(pi-varphi)` (c) 0 (d) `2cotvarphi`

Options

A. `-2tanphi`
B. `-2tan(pi-phi)`
C. 0
D. `2cotphi`

Answer

A

Explanation

(1) The slope of tangent at P is `1//t_(1)`
and that at Q is `1//t_(2)`. Therefore,
`cottheta_(1)=t_(1)andcottheta_(2)=t_(2)`
Slope of PQ `=(2)/(t_(1)+t_(2))`
`:." Slope of OR "-(t_(1)+t_(2))/(2)=tanphi`
(Angle in a semicircle is `90^(@))`
`ortanphi=-(1)/(2)(cottheta_(1)+cottheta_(2))`
`:.cottheta_(1)+cottheta_(2)=-tanphi`