Question

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

Solution

Correct option is

(x – 2h)2 – (y – 2k)2 = a2

Let A, B, C and D have coordinate (xiyi), i = 1, 2, 3, 4, then

                             ….. (1)

And                       ….. (2)

Let the centroid of ΔABC be  , then 

 .     

So, from (1) and (2),

      

But (x4y4) lies on x2 – y2 = 9a2

So,

 

 is (– 2h)2 – (y – 2k)2 = a2     

SIMILAR QUESTIONS

Q1

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

Q2

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 (x1y1). Then  is equal to

Q3

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

Q4

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

Q5

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

Q6

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 –

Q7

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 –

Q8

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 – 

Q9

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

Q10

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 –