Given the base BC of the triangle ABC and if `angleC-angleB=k,` a constant, show that the locus of the vertex A is a hyperbola.

A (-1,0) and B (2,0) are two given points. A point M is moving in such a way that the angle B in the triangle AMB remains twice as large as the angle A. Show that the locus of the point M is a hyperbola. Fnd the eccentricity of the hyperbola.

Having given the bases and the sum of the areas of a number of triangles which have a common vertex, show that the locus of the vertex is a straight line.

Given the base of a triangle and the ratio of the tangent of half the base angles .Show that the vertex moves on a hyperbola whose foci are the extremities of a diameter

If on a given base B C , a triangle is described such that the sum of the tangents of the base angles is m , then prove that the locus of the opposite vertex A is a parabola.

Statement 1 : Given the base B C of the triangle and the radius ratio of the excircle opposite to the angles Ba n dCdot Then the locus of the vertex A is a hyperbola. Statement 2 : |S P-S P|=2a , where Sa n dS ' are the two foci 2a is the length of the transvers axis, and P is any point on the hyperbola.

If the base of a triangle and the ratio of tangent of half of base angles are given, then identify the locus of the opposite vertex.

If the base of a triangle and the ratio of tangent of half of base angles are given, then identify the locus of the opposite vertex.

In figure, angleB lt angleA and angleC lt angleD . Show that AD lt BC .

Given two points A and B. If area of triangle ABC is constant then locus of point C in space is

The base AB of a triangle is fixed and its vertex C moves such that sin A =k sin B (kne1) . Show that the locus of C is a circle whose centre lies on the line AB and whose radius is equal to (ak)/((1-k^2)) , a being the length of the base AB.