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Show that the locus of the midpoints of chords of the prabolay^{2}=4ax\, which touch the circle x^{2}+y^{2}=a^{2}\, is (y^{2}-2ax)^{2}=a^{2}(y^{2}+4a^{2})\,

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

Equation of the chord is




This line touch the circle with centre (0,0) and radius a.

The perpendicualar distance from the centre is equal to the readius,


{\frac  {|y_{1}^{{2}}-2ax_{1}|}{{\sqrt  {y_{1}^{{2}}+4a^{2}}}}}=a\,

Squaring on bothsides,we get


Hence the locus of P is


Main Page:Geometry:The Parabola