Tangents and Normals

Tangents & Normals Sheet – 1 : If the circles touch each other show that . Find the equation of a circle which is concentric with circle and touches the line . Find the equation of the tangents to the circle drawn through the point of intersection of the lines . Show that the straight line touches the circle and find the point of contact. Find the common tangents to the circles Prove that the locus of a point, the tangents from which to two intersecting circles are equal, is the common chord of the circles. Find the equation of the tangent to which is parallel to . A tangent to the parabola makes an angle with the positive direction of the x-axis. Find its equation and the point of contact. Prove that the length of any tangent to a parabola intercepted between its point of contact and the directrix subtends a right angle at the focus. Find the equation of the normal to the parabola which makes with the x-axis. Find also the foot of the normal. Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola is . Two tangents to the parabola meet at an angle of ; prove that the locus of their point of intersection is the curve. Find the locus of the point of intersection of normals at the extremities of any focal chord of the parabola . Two equal parabolas have the same vertex, and their axes are at right angles; prove that their common tangent touches each at an end of its latus rectum. Find the equations to the tangents to the ellipse which are parallel to the line . Find also the point of contact. The normal at P to the ellipse meets the axes in G and H; find the loci of the middle points of PG and GH. Prove that the portion of the tangent at any point of a hyperbola intercepted between the asymptotes is bisected at the point of contact. Show that the area of the triangle cut off from the asymptotes by a tangent to the hyperbola is constant. Find the equation of the circle having its centre at the point (3, 4) and touching the straight line . Find the common tangents of the following pair of circles: . Find the length of the tangents drawn from the point (7, 2) to the circle . Show that the lengths of the tangents from any point on the line to the circles and are equal. Show that the length of the tangent drawn from any point on the circle to the circle is . Find out the equations of the tangents to the parabola at the extremity of the latus rectum . Show that the line touches the parabola if . Show that the line touches the parabola for all values of . Find the length of the normal chord of the parabola through an extremity of the latus rectum. Show that the point of intersection of the tangents to the parabola at the points and is . Show that the point of intersection of normals to the parabola at the points and is If the normal at the point to the parabola meets the parabola again at point , show that . If the normals at the points and to the parabola meet on the parabola, prove that . The two tangents drawn from a point P to the parabola are at right angles. Find the locus of P. 2 -an endeavour of Dhruba Chakraborty