# Problems on Circle

We will learn how to solve different types of problems on circle.

1. Find the equation of a circle of radius 5 whose centre lies on x-axis and passes through the point (2, 3).

Solution:

Let the coordinates of the centre of the required circle be C(a, 0). Since it passes through the point P(2, 3).

⇒ CP = 5

⇒ $$\mathrm{\sqrt{(a - 2)^{2} + (0 - 3)^{2}}}$$ = 5

⇒ (a - 2)$$^{2}$$ + 9 = 25

⇒ (a - 2)$$^{2}$$ = 25 - 9

⇒ (a - 2)$$^{2}$$ = 16

⇒ a - 2 = ± 4

⇒ a = -2 or 6

Thus, the coordinates of the centre are (-2, 0) and (6, 0).

Hence, the equation of the required circle are

(x - 2)$$^{2}$$ + (y – 0)^2 = 5^2 and (x – 6)$$^{2}$$ + (y – 0)$$^{2}$$ = 5$$^{2}$$

⇒ x$$^{2}$$ + y$$^{2}$$ + 4x – 21 = 0 and x$$^{2}$$ + y$$^{2}$$ – 12x + 11 = 0

2. Find the equation of the circle which passes through the points (3, 4) and (- 1, 2) and whose centre lies on the line x - y = 4.

Solution:

Let the equation of the required circle be

x$$^{2}$$ + y$$^{2}$$ + 2gx + 2fy + c = 0 ............... (i)

According to the problem the equation (i) passes through the points (3, 4) and (- 1, 2). Therefore,

9 + 16 + 6g + 8f + c = 0 ⇒ 6g + 8f + c = - 25 ............... (ii)

and 1 + 4 - 2g + 4f + c = 0 ⇒ - 2g + 4f + c = - 5 ............... (iii)

Again according to the problem, the centre of the circle (i) lies on the line x - y = 4.

Therefore,

- g  - (- f) = 4

⇒ - g + f = 4 ............... (iv)

Now, subtract the equation (iii) from (ii) we get,

8g + 4f = - 20

⇒ 2g + f = - 5 ............... (v)

Solving equations (iv) and (v) we get, g = - 3 and f = 1.

Putting g = - 3 and f = 1 in (iii) we get, c = -15.

Therefore, the equation of the requited circle is x$$^{2}$$ + y$$^{2}$$ - 6x + 2y - 15 = 0.

More problems on circle:

3. Find the equation to the circle described on the common chord of the given circles x$$^{2}$$ + y$$^{2}$$ - 4x - 5 = 0 and x$$^{2}$$ + y$$^{2}$$ + 8x + 7 = 0 as diameter.

Solution:

Let, S$$_{1}$$ = x$$^{2}$$ + y$$^{2}$$ - 4x - 5 = 0 ............... (i)

and S$$_{2}$$ = x$$^{2}$$ + y$$^{2}$$ + 8x + 7 = 0 ............... (ii)

Then, the equation of the common chord of the circles (1) and (2) is,

S$$_{2}$$ - S$$_{1}$$ = 0

⇒ 12x + 12 = 0

⇒ x + 1 = 0 ............... (iii)

Let the equation of the circle described on the common chord of (i) and (ii) as diameter be

x$$^{2}$$ + y$$^{2}$$  - 4x - 5 + k(x + 1) = 0

⇒ x$$^{2}$$ + y$$^{2}$$  - (4 - k)x - 5 + k = 0 ............... (iv)

Clearly, the co-ordinates of the centre of the circle (4) are ($$\frac{4 - k}{2}$$, 0) Since the common chord (iii) is a diameter of the circle (iv) hence,

$$\frac{4 - k}{2}$$ + 1 = 0

⇒ k = 6.

Now putting the value of k = 6 in x$$^{2}$$ + y$$^{2}$$ - (4 - k) x- 5 + k = 0 we get,

x$$^{2}$$ + y$$^{2}$$  - (4 - 6) x - 5 + 6 = 0

⇒ x$$^{2}$$ + y$$^{2}$$ + 2x + 1 = 0, which is the required equation of the circle.