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Let Gamma be a circle with diameter AB and centre O.Let l be the tangent toGamma at B.For each point M onGamma 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 Gammais |
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Answer» an ARC of a circle  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 |
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