The Angle Between Lines Joining Origin To The Points Of Intersection Of The Line  and The Curve y2 – x2 = 4 Is

Two straight lines pass through the fixed points  and have gradients whose product is k > 0. Show that the locus of the points of intersection of the lines is a hyperbola.

Find the equation of the triangles drawn from the point (–2, –1) to the hyperbola 2x² – 3y² = 6.

Find the range of ‘a’ for which two perpendicular tangents can be drawn to the hyperbola from any point outside the hyperbola

.

Find the hyperbola whose asymptotes are 2x – y = 3 and 3x + y – 7 = 0 and which passes through the point (1, 1).

The locus of a variable point whose distance from (–2, 0) is  times its distance from the line , is

A variable straight line of slope 4 intersects the hyperbola xy = 1 at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is

Let P and , where , be two points on the hyperbola . If (h, k) is the point of intersection of the normal’s at P and Q, then k is equal to

If x = 9 is the chord of contact of the hyperbola x² – y² = 9, then equation of corresponding of tangents is

The locus of the mid-point of the chord of the circle x² + y² = 16, which are tangent to the hyperbola 9x² – 16y² = 144 is

If a circle cuts a rectangular hyperbola xy = c² in A, B, C, D and the parameters of these four points be t₁, t₂, t₃ and t₄ respectively. Then