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T is a point on the tangent to a parabola `y^2= 4ax` at its point `P. TL and TN` are the perpendiculars on the focal radius SP and the directrix of the parabola respectively. Then (A)SL=2(TN) (B) 3(SL)=2(TN) (C) SL=(TN) (D) 2(SL)=3(TN) |
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Answer» `y^2=4ax` Tangent at `P:x+at^2` Let`T(x_1,y_1)` lies on tangent `ty_1=x_1+at^2` `m_(SP)=(2at-0)/(at^2-a)-(2t)/(t^2-1)` `m_(sp)*m_(tl)=-1` `(2t)/(t^2-1)*m_(tl)=-1` `m_(tl)=(1-t^2)/(2t)` Equation of TL `y-y_1=(1-t^2)/(2t)*(x-x_1)` SL is perpendicular on line TC `D_1=SL=|((-2ty_1+a(t^2-1)-x_1(t^2-1))/(sqrt(4t^2+(t^2-1)^2)))|` solving this `SL=x_1+a` perpendicular distance `D_2=TN=(x_1+a)/(sqrt(1^2+0^2))=x_1+a` `SL+TN+x_1+a` |
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