**Exercise 11.1**

**In each of the following Exercises 1 to 5, find the equation of the circle with:**

**1. Centre (0, 2) and radius 2.**

**Ans. **Given: and

Equation of the circle =

**2. Centre **** and radius 4.**

**Ans. **Given: and

Equation of the circle =

**3. Centre **** and radius **

**Ans. **Given: and

Equation of the circle =

**4. Centre **** and radius **

**Ans. **Given: and

Equation of the circle:

**5. Centre **** and radius **

**Ans. **Given: and

Equation of the circle =

**In each of the following Exercises 6 to 9, find the centre and radius of the circles.**

**6. **

**Ans. **Given: Equation of the circle:

……….(i)

On comparing eq. (i) with

and

**7. **

**Ans. **Given: Equation of the circle:

……….(i)

On comparing eq. (i) with

and

**8. **

**Ans. **Given: Equation of the circle:

……….(i)

On comparing eq. (i) with

and

**9. **

**Ans. **Given: Equation of the circle:

……….(i)

On comparing eq. (i) with

and

**10. Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line **

**Ans. **The equation of the circle is ……….(i)

Circle passes through point (4, 1)

……….(ii)

Again Circle passes through point (6, 5)

……….(iii)

From eq. (ii) and (iii), we have

……….(iv)

Since the centre of the circle lies on the line

……….(v)

On solving eq. (iv) and (v), we have

Putting the values of and in eq. (ii), we have

Therefore, the equation of the required circle is

**11. Find the equation of the circle passing through the points (2, 3) and **** and whose centre is on the line **

**Ans. **The equation of the circle is ……….(i)

Circle passes through point (2, 3)

……….(ii)

Again Circle passes through point (–1, 1)

……….(iii)

From eq. (ii) and (iii), we have

……….(iv)

Since the centre of the circle lies on the line

……….(v)

On solving eq. (iv) and (v), we have

Putting the values of and in eq. (ii), we have

Therefore, the equation of the required circle is

**12. Find the equation of the circle with radius 5 whose centre lies on ****axis and passes through the point (2, 3).**

**Ans. **Since the centre of circle lies on axis, therefore the coordinates of centre is

Now the circle passes through the point (2, 3). According to the question,

or

Taking , Equation of the circle is

Taking , Equation of the circle is

**13. Find the equation of the circle passing through (0, 0) and making intercept **** and **** on the coordinate axes.**

**Ans. **The circle makes intercepts with axis and with axis.

OA = and OB =

Coordinates of A and B are and respectively.

Now the circle passes through the points O (0, 0), A and B.

Putting these coordinates of three points in the equation of the circle,

………(i)

And

And

Putting the values of and in eq. (i), we have

**14. Find the equation of the circle with centre (2, 2) and passes through the point (4, 5).**

**Ans. **The equation of the circle is ……….(i)

Since the circle passes through point (4, 5) and coordinates of centre are (2, 2).

Radius of circle = = =

Therefore, the equation of the required circle is

**15. Does the point **** lie inside, outside or on the circle **** ?**

**Ans. **Given: Equation of the circle

On comparing with , we have and

Now distance of the point from the centre (0, 0)

= = = = 4.3 <5

Therefore, the point lies inside the circle.

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