Question

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

Solution

Correct option is

xy = 2c2

x – a = 0                                                        ……. (1)

y – b =  0                                                       ……. (2)

are conjugate w.r.t.

Let pole of (1) be (x1y1) w.r.t.                       ....... (3)

Then its polar is x1y + xy= 2c2                      …… (4)

Compare (1) and (4)

        

Now line (1) and (2) are conjugate and so pole of one line lies on the other line

⇒  (x1y1) lies on (2)         ⇒ y1 – b =0

∴ locus of (a, b) is xy = 2c2

SIMILAR QUESTIONS

Q1

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

Q2

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

Q3

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

Q4

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 =

Q5

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

Q6

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

Q7

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

Q8

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

Q9

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

Q10

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