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

The perpendicular from the origin to the tangent at any point on a curve is equal to the abscissa of the point of contact. Also curve passes through the point (1,1). Then the length of intercept of the curve on the x-axis is__________

The perpendicular from the origin to the tangent at any point on a curve is equal to the abscissa of the point of contact. Also curve passes through the point (1,1). Then the length of intercept of the curve on the x-axis is__________

The slope of the tangent any point on a curve is lambda times the slope of the joining the point of contact to the origin. Formulate the differential equation and hence find the equation of the curve.

Fill in the blanks: The common point of a tangent and the circle is called...... A circle may have ..... parallel tangents. A tangent to a circle intersects it in ..... point(s). A line intersecting a circle in two points is called a ........ (v) The angle between tangent at a point on a circle and the radius through the point is .........

. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Prove that tangent drawn at the mid point of the are of a circle is pallelar to the chord joing the ends of point of the are

In both an ellipse and hyperbola , prove that the focal distance of any point and the perpendicular from the centre upon the tangent at it meet on a circle whose centre is the focus and whose radius is the semi-transverse axis.

Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of intersection of its diagonals.

Prove that the circle drawn with any side of a rhombus as a diameter, passes through the point of intersection of its diagonals.

Perpendiculars are drawn, respectively, from the points Pa n dQ to the chords of contact of the points Qa n dP with respect to a circle. Prove that the ratio of the lengths of perpendiculars is equal to the ratio of the distances of the points Pa n dQ from the center of the circles.