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Show that the locus of the midpoints of chords of the parabola y^{2}=6x\, and which touch the circle x^{2}+y^{2}+4x-12=0\, is [y^{2}-3x-6]^{2}=16(y^{2}+9)\,

Equation of the midpoints of chords of the given parabola if P(x_{1},y_{1})\, is the midpoint,



This line touches the circle with centre (-2,0)\, and radius is 4.

Hence the distance from the centre to the line is

{\frac  {|6-y_{1}^{{2}}+3x_{1}|}{{\sqrt  {y_{1}^{{2}}+9}}}}=4\,

Squaring on bothsides,we get


Therefore,the locus is


Main Page:Geometry:The Parabola