Geo5.1.35

From Example Problems
Jump to: navigation, search

Show that the locus of poles of chords of the parabolay^{2}=4ax\, at a constant distance b from the vertex is b^{2}y^{2}+4a^{2}(b^{2}-x^{2})=0\,

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

Equation to the polar of P w.r.t the parabola is

yy_{1}-2a(x+x_{1})=0\,

Given perpendicular distance of this line from S(0,0)\, is b.

Hence

{\frac  {|-2a(0+x_{1})|}{{\sqrt  {y^{2}+4a^{2}}}}}=b\,

Squaring on bothsides,

4a^{2}x_{1}^{{2}}=b^{2}(y_{1}^{{2}}+4a^{2})\,

-b^{2}y_{1}^{{2}}-4a^{2}(b^{2}-x_{1}^{{2}})=0\,

b^{2}y_{1}^{{2}}+4a^{2}(b^{2}-x_{1}^{{2}})=0\,

Therefore,the locus of P is

b^{2}y^{2}+4a^{2}(b^{2}-x^{2})=0\,

Main Page:Geometry:The Parabola