Prove that the tangent and normal at any point of an ellipse bisect the external and internal angles between the focal distances of the point

If (-2,5) and (3,7) are the points of intersection of the tangent and normal at a point on a parabola with the axis of the parabola, then the focal distance of that point is

What is the sum of focal radii of any point on an ellipse equal to ?

If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Tangent Normal (Angle between Curves)

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segments joining the points of contact at the centre.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Prove that the tangent at any point of circle is perpendicular to the radius through the point of contact.

If the tangent at any point of an ellipse x^2/a^2 + y^2/b^2 =1 makes an angle alpha with the major axis and an angle beta with the focal radius of the point of contact then show that the eccentricity ‘e’ of the ellipse is given by the absolute value of cos beta/cos alpha