Geo5.3.28

From Example Problems
Jump to: navigation, search

Show that the locus of poles w.r.t parabola y^{2}=4ax\, of the tangents to the hyperbola x^{2}-y^{2}=a^{2}\, is the ellipse 4x^{2}+y^{2}=4a^{2}\,

The equation of polar P w.r.t the parabola given is

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

This line is a tangent to hyperbola x^{2}-y^{2}=a^{2}\,

The condition of tangency is a^{2}l^{2}-b^{2}m^{2}=n^{2}\,

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

4a^{4}-a^{2}y_{1}^{{2}}=4a^{2}x_{1}^{{2}}\,

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

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

Hence the locus of P is 4x^{2}+y^{2}=4a^{2}\,


Main Page:Geometry:Hyperbola