Geo5.1.33

From Example Problems
Jump to: navigation, search

The chord of contact of tangents from a point P to the parabola y^{2}=4ax\, touches the circle x^{2}+y^{2}=b^{2}\,.Prove that the locus of P is4a^{2}x^{2}=b^{2}(y^{2}+4a^{2})\,

Let P(x_{1},y_{1})\, be a point.

From the given parabola,

Equation of the chord of contact is

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

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

If this line touches the circle x^{2}+y^{2}=b^{2}\,

The perpendicular distance from the centre of the circle to the line equal to the radius,

Therefore,

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

2ax=b[{\sqrt  {4a^{2}+y_{1}^{{2}}}}]\,

Squaring on bothsides,we have

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

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

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

Hence the locus of P is

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


Main Page:Geometry:The Parabola