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  • The locus of the middle points of all chords of a parabola , which are drawn t

The locus of the middle points of all chords of a parabola \mathrm{y^2=4 a x}, which are drawn through the vertex is

Option: 1

\mathrm{y^2=2 a x}


Option: 2

\mathrm{y^2=a x}


Option: 3

\mathrm{y^2=2 a x+4}


Option: 4

\mathrm{y^2=-2 a x}


Answers (1)

Let \mathrm{P\left(x_1, y_1\right)} be the middle point of a chord \mathrm{O A}. The equation of \mathrm{O A} is

                  \mathrm{ T=S_1 }

Or  \mathrm{y y_1-2 a\left(x+x_1\right)=y_1^2-4 a x_1}

It passes through \mathrm{O(0,0)}. The condition is

\mathrm{ 0-0-2 a x_1=y_1^2-4 a x_1 }
The locus of  \mathrm{\left(x_1, y_1\right)} is \mathrm{y^2-2 a x=0.}

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Kshitij

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