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  • If the locus of a point that divides a chord of slope 2 parabola internally in the

If the locus of a point that divides a chord of slope 2 parabola \mathrm{y^2=4 x} internally in the ratio 1: 2 is parabola. Th vertex is
 

Option: 1

(1 / 9,8 / 9)
 


Option: 2

(2 / 9,8 / 9)
 


Option: 3

(3 / 9,7 / 9)
 


Option: 4

(4 / 9,8 / 9)


Answers (1)

best_answer

\mathrm{Let \: P Q\text{ be the chord of the parabola}\: \: y^2=4 x}

\mathrm{Let \: P\left(t_1\right), Q\left(t_2\right) \text{be the points then slope of }P Q=\frac{2}{t_1+t_2}=2}

\mathrm{ \Rightarrow t_1+t_2=1 }    ........(1)

\mathrm{ \text{The point which divides} \: P Q\:\text{ in the ratio} 1: 2\: is }

\mathrm{ R=\left(\frac{2 t_1^2+t_2^2}{3}, \frac{4 t_1+2 t_2}{3}\right)=(x, y) }

\mathrm{ \Rightarrow 3 x=2 t_1{ }^2+t_2{ }^2, 3 y=4 t_1+2 t_2 }       ........(2)

\mathrm{ Eliminating\: t_1 \: and \: t_2\: \text{using (1) and (2), we get} }

\mathrm{ 4 x=9 y^2-16 y+8 }

\mathrm{ \text { or }\left(y-\frac{8}{9}\right)^2=\frac{4}{9}\left(x-\frac{2}{9}\right) }

\mathrm{\text{which a parabola whose vertex is} \left(\frac{2}{9}, \frac{8}{9}\right)}




 

Posted by

Ritika Harsh

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