A, B, C And D Are The Points Of Intersection Of A Circle And A Rectangular Hyperbola Which Have Different Centers. If AB Passes Through The Center Of The Hyperbola, Then CD Passes Through

The equation of normal to the rectangular hyperbola xy = 4 at the point P on the hyperbola which is parallel to the line

2x – y = 5 is

A straight line intersects the same branch of the hyperbola  in P₁ and P₂ and meets its asymptotes in Q₁ and Q₂. Then P₁Q₂ – P₂Q₁ is equal to

From a point on the line y = x + c, c (parameter), tangents are drawn to the hyperbola  such that chords of contact pass through a fixed point (x₁, y₁). Then  is equal to

If the portion of the asymptotes between center and the tangent at the vertex of hyperbola  in the third quadrant is cut by the line  being parameter, then

Five points are selected on a circle of radius a. the centers of the rectangular hyperbolas, each passing through four of these pints lie on a circle of a radius

Sa circle with fixed center (3h, 3k) and of variable radius cuts the rectangular hyperbola x² – y² = 9a² at the points A, B, C, D. The locus of the centroid of the triangle ABC is given by

If PQ is a double ordinate of the hyperbola  such the OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola satisfies –

A rectangular hyperbola passes through the points A(1, 1), B(1, 5) and C(3, 1). The equation of normal to the hyperbola at A(1, 1) is –

If a variable line  which is a chord of the hyperbola  subtends a right angle at the centre of the hyperbola then it always touches a fixed circle whose radius is – 

If values of m for which the line  touches the hyperbola 16x² – 9y² = 144 are the roots of the equation x² –(a + b)x – 4 = 0, then value of (a + b) is equal to –