Geo5.1.26

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If a chord of the parabola y^{2}=4ax\, touches the parabola y^{2}=4bx\,. Show that the tangents at its extremities meet on the parabola by^{2}=4a^{2}x\,

Let QR be a chord of the parabola y^{2}=4bx\,

Let P(x_{1},y_{1})\, be the point of intersection of the tangent at Q and R to the above equation.

Then QR is the chord of contact of P w.r.t the parabola.

Therefore,the equation of QR is yy_{1}-2a(x+x_{1})=0\,

2ax-yy_{1}+2ax_{1}=0\,

Given that this equation touches the parabola y^{2}=4bx\,

2a(2ax_{1})=by_{1}^{{2}}\,

4a^{2}x_{1}=by_{1}^{{2}}\,

Therefore,the locus of P is by^{2}=4a^{2}x\,


Main Page:Geometry:Plane|The Parabola