Prove that in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

The perpendicular from the origin to a line meets it at the point (2,9) , find the equation of the line.

The perpendicular from the origin to a line meets it at the point (-2, 9) find the equation of the line.

Prove that the tangent and the radius through the point of contact of a circle are perpendicular to each other.

The locus of the foot of perpendicular from my focus of a hyperbola upon any tangent to the hyperbola is the auxiliary circle of the hyperbola. Consider the foci of a hyperbola as (-3, -2) and (5,6) and the foot of perpendicular from the focus (5, 6) upon a tangent to the hyperbola as (2, 5). The directrix of the hyperbola corresponding to the focus (5, 6) is

Prove that if two tangents are drawn to a circle from a point outside it, then the line segments joining the point of contacts and the exterior point are equal.

Show that , The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Prove analytically that in any triangle the perpendiculars drawn from the vertices upon the opposite sides are concurrent.

Prove that if two tangents are drawn to a circle from a point outside it, then the line segments joining the point of contact and the exterior point are equal and they subtend equal angles at the centre.

Prove that the straight line joining the upper end of one latus rectum and lower end of other latus rectum of an ellipse passes through the centre of the ellipse .

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