Question

Locus of mid-point of the chords of contact of x2 + y2 = 2 from the points on the line 3x + 4y = 10 is a circle with centre P. If O be the origin then OP is equal to

Solution

Correct option is

1/2

 

Chord with mid-point (hk) is 

hx + ky = h2 + k2                                     …(1)

chord of contact of (x­1y1) is

xx1 + yy1 = 2                                            …(2)

Comparing, we get

(x1y1) lies on 3x + 4y = 10 ⇒ 6h + 8k = 10(h2 + k2)

∴ locus of (hk) is   

  

∴ OP = 1/2.

SIMILAR QUESTIONS

Q1

Find the equation of circle which cuts the circle x2 + y2 + 5x + 7y + 4 = 0 orthogonally, has its centre on the line x = 2, and passes through the point (4, –1).

Q2

Find the point of intersection of the line 2x + 3y = 18 and the circle x2 +y2 = 25.

Q3

Find the equation of the normal to the circle x2 + y2 – 5x + 2y – 48 = 0 at point (5, 6).

Q4

Find the equation of the tangent to the circle x2 + y2 = 16 drawn from the point (1, 4).

Q5

Find the equation of the image of the circle x2 + y2 + 16x – 24y + 183 = 0 by the line mirror 4x + 7y + 13 = 0.

Q6

Find the equation of the normal to the circle x2 + y2 = 2x, which is parallel to the line x + 2y = 3.

Q7

 

Find the equation of the circle which cuts orthogonally each of the three circles given below: 

x2 + y2 – 2x + 3y – 7 = 0, x2 + y2 + 5x – 5y + 9 = 0 and x2 + y2 + 7x – 9x + 29 = 0.

Q8

Circum centre of the triangle PT1T2 is at

Q9

If P is taken to be at (h, 0) such that P’ lies on the circle, the area of the rhombus is

Q10

Suppose ax + bx + c = 0, where abc are in A.P. be normal to a family or circles. The equation of the circle of the family which intersects the circle x2 + y2 – 4x – 4y – 1 = 0 orthogonally is