Question

If a rectangular hyperbola whose center is C, is cut by any circle of radiusr in the four points P, Q, R, S, then 

CP2 + CQ2 + CR2 + CS2 =

Solution

Correct option is

4r2

Circle is x2 + y2 + 2gx + 2fy + d = 0              ……. (1)

Its radius = r ⇒ r2 = g2 + f2 – d                     …… (2)

Rectangular hyperbola as xy = c2                 …… (3)

Whose center is at (0, 0). Take , a point on (2).

Put this in (1).

⇒ c2t4 + 2gct3 + 2fct + dt2 + c= 0                 …… (3)

(3) Gives point of intersections

             

Now   

      

Here  

       

              .     by   (3)

          ….. (4)

Similarly if we replace 

                                       ….. (5)

  Add (4) and (5)

SIMILAR QUESTIONS

Q1

The line 3x + 2y + 1 = 0 meets the hyperbola 4x2 – y2 = 4a2 in the points P and Q. The coordinates of point intersection of the tangents at and Qare

Q2

The eccentricity of the hyperbola whose latus rectum is half of its transverse axis is

Q3

The number of tangents to the hyperbola  through (4, 3) is

Q4

The equation of the hyperbola referred to it axes as axes of coordinates whose latus rectum is 4 and eccentricity is 3, is

Q5

If θ is the angle between the asymptotes of the hyperbola   with eccentricity e, then 

Q6

If the two lines x – a = 0 and y – b =  0 are conjugate w.r.t. the hyperbolaxy = c2, then the locus of (a, b) is

Q7

The equation of the tangents to the conic 3x2 – y2 = 3 perpendicular to the line x + 3y = 2 is

Q8

If P is a point on the hyperbola 16x– 9y2 = 144 whose foci are S1 andS2, then PS1 – PS2 =

Q9

The length of the transverse axis of a hyperbola is 7 and it passes through the point (5, –2). The equation of the hyperbola is

Q10

The locus of the point of intersection of the lines (x + y)t = a and x – y = at, where t is the parameter, is