Question

Find the locus of the pole of the line lx + my + n = 0 with respect to the circle which touches y-axis at the origin.

Solution

Correct option is

y(lx – n) = mx2

 

Let centre of circle be C(h, 0) since circle touches y-axis at the origin.

∴        Radius of the circle = h

∴ Equation of the circle 

                (x – h)2 + (y – 0)2 = h2

Let the pole be (x1y1

∴ Equation of the polar w.r.t circle (1) is 

                 xx1 + yy1 – h (x + x1) = 0

And given line lx + my + n = 0 which is (2):

  

From first two relations, we get

                                 

and from last two relations, we get

                                 

   

  

   

∴ Locus of pole is          y(lx – n) = mx2      

 

 

 
 

 

 

SIMILAR QUESTIONS

Q1

Show that the locus of the point, the powers of which with respect to two given circles are equal, is a straight line.

Q2

 

Find the condition that chord of contact of any external point

(hk) to the circle x2 + y2 = a2 should subtend right angle at the centre of the circle.

Q3

The chord of contact of tangents drawn from a point on the circle x2 +y2 = a2 to the circle x2 + y2 = b2 touches the circle x2 = y2 = c2. Show that abc are in GP.

Q4

Find the equation of the chord x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (–2, 4).

Q5

Find the middle point of the chord intercepted on line lx + my + n = 0 by the circle x2 + y2 = a2.

Q6

Find the locus of middle points of chords of the circle x2 + y2 = a2, which subtend right angle at the point (c, 0).

Q7

Find the equations of the tangents from the point A(3, 2) to the circle x2y2 + 4x + 6y + 8 = 0 .

Q8

If two tangents are drawn from a point on the circle x2 + y2 = 50 to the circle x2 + y2 = 25 then find the angle between the tangents.

Q9

 

Find the equation of the diameter of the circle

x2 + y2 + 2gx + 2fy + c = 0 which corresponds to the chord ax = by + d= 0.

Q10

Examine if the two circles x2 + y2 – 2x – 4y = 0 and x2 + y2 – 8y – 4 = 0 touch each other externally or internally.