If the product of the perpendiculars from the foci upon the polar of P be constant and equal to ` c^(2)` prove that the locus of P is the ellipse ` b^(4) x^(2) ( c^(2) + a^(2) e^(2)) + c^(2) a^(4) y^(2) = a^(4) b^(4)`

Product of perpendiculars drawn from the foci upon any tangent to the ellipse 3x^(2)+4y^(2)=12 is

In /_ABC, if c^(4)-2(a^(2)+b^(2))c^(2)+a^(4)+a^(2)b^(2)+b^(4)=0 then

A point P moves such that the chord of contact of the pair of tangents from P on the parabola y^(2)=4ax touches the rectangular hyperbola x^(2)-y^(2)=c^(2). Show that the locus of P is the ellipse (x^(2))/(c^(2))+(y^(2))/((2a)^(2))=1

A point P moves such that the chord of contact of the pair of tangents P on the parabola y^(2)=4ax touches the the rectangular hyperbola x^(2)-y^(2)=c^(2). Show that the locus of P is the ellipse (x^(2))/(c^(2))+(y^(2))/((2a)^(2))=1

Prove that the product of the perpendiculars from the foci upon any tangent to the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is b^(2)

If a/b = c/d =e/f = 4, then (m ^(2) a + n^(2) c + p ^(2) e )/( m ^(2) b + n ^(2) d + p ^(2) f )=?