Question

A circle has radius 3 units and its centre lies on the line y = x – 1. Find the equation of the circle if it passes through (7, 3).

Solution

Correct option is

x2 + y2 – 14x – 12y + 76 = 0 & x2 + y2 – 8x – 6y + 16 = 0

 

Let the centre of the circle be (hk). since the centre lies on y = x – 1, we get

                                      k = h – 1                  …(1)

Since the circle passes through the point (7, 3), therefore the distance of the centre from this point is the radius r of the circle. we have

                                  

     … from (1)

⇒                           h2 – 11h + 28 = 0

or                           (h – 7) (h – 4) = 0

or                           h = 7 and h = 4

from h = 7, we get k = 6 from (1)

and for h = 4, we get k = 3 from (1).

Hence there are two circles which satisfy the given conditions. They are 

         (x – 7)2 + (y – 6)2 = 9      or    x2 + y2 – 14x – 12y + 76 = 0

and   (x – 4)2 + (y – 3)2 = 9      or    x2 + y2 – 8x – 6y + 16 = 0.

SIMILAR QUESTIONS

Q1

Circum centre of the triangle PT1T2 is at

Q2

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

Q3

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

Q4

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

Q5

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

Q6

Find the equation of that chord of the x2 + y2 = 15 which is bisected at (3, 2).

Q7

 

Find the centre and radius of the circle 

             2x2 + 2y2 = 3x – 5y + 7

Q8

Find the equation of the circle whose centre is the point of intersection of the lines 2x – 3y + 4 = 0 and 3x + 4y – 5 = 0 and passes through the origin.

Q9

Find the equation of the circle concentric with the circle x2 + y2 – 8x + 6y– 5 = 0 and passing through the point (–2, –7).

Q10

 

Find the area of an equilateral triangle inscribed in the circle

                       x2 + y2 + 2gx + 2fy + c = 0