Geo5.1.39

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A tangent to the parabola y^{2}+4bx=0\, meets y^{2}=4ax\, at P and Q. Prove that the locus of the midpoint of PQ is y^{2}(2a+b)=4a^{2}x\,

Let M(x_{1},y_{1})\, be the midpoint of the chord PQ of the given parabola

Equation to chord is yy_{1}-2a(x+x_{1})=y_{1}^{{2}}-4ax_{1}\,

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

The line is a tangent to y^{2}=4(-bx)\,

2a(y_{1}^{{2}}-2ax_{1})=(-b)(-y_{1})^{{2}}\,

Therefore,the locus of M is

2ay^{2}-4a^{2}x=-by^{2}\,

y^{2}(2a+b)=4a^{2}x\,


Main Page:Geometry:The Parabola