Let `Gamma` be a circle with diameter AB and centre O.Let l be the tangent to`Gamma` at B.For each point M on`Gamma` different from A,consider the tangent t at M and let it interest l at P.Draw a line parallel to AB through P intersecting OM at Q.The locus of Q as M varies over `Gamma`is

Options

A. an arc of a circle
B. a parabola
C. an arc of an ellipse
D. a branch of a hyperbola

Answer

B

Explanation

Equation of tangent at M, `x cos theta + y sin theta =r`
put x=r, to get y-coordinate of point P.
`r cos theta + y sin theta =r`
`implies y=(r(1-cos theta))/(sin theta)=(r .2.sin^(2)"" theta/2)/(2.sin ""theta/2.cos"" theta/2)=r tan ""theta/2`
`:. P=(r,rtan""(theta)/(2))`
`:.` Q has y-coordinate same as point P
`:. K= r tan""theta/2 impliestan""(theta)/(2)=(K)/(r)`
Slope of tangent at `M= - cot theta`
Slope of `OQ=(K)/(h)`
`:. (K)/(h),(- cot theta)= -h implies than theta =(K)/(h)`
`implies (2 tan""(theta)/(2))/(1- tan^(2)""(theta)/(2))=(K)/(h ) implies(2. (K)/(r))/(1-(K^(2))/(r^(2)))=(K)/(h)`
`implies (2h)/(r)=1-(K^(2))/(r^(2)) implies(2h)/(r)=(r^(2)-K^(2))/(r^(2))`
`implies 2hr=r^(2)-K^(2)`
`implies y^(2)=r^(2)-2Kr`
`y^(2)= - 2r(x-r//2)`
`:.` Parabola