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).

Therefore, CP = radius

⇒ 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.

 The Circle




11 and 12 Grade Math 

From Problems on Circle to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



Share this page: What’s this?

Recent Articles

  1. Method of L.C.M. | Finding L.C.M. | Smallest Common Multiple | Common

    Apr 15, 24 01:29 AM

    LCM of 24 and 30
    We will discuss here about the method of l.c.m. (least common multiple). Let us consider the numbers 8, 12 and 16. Multiples of 8 are → 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, ......

    Read More

  2. Common Multiples | How to Find Common Multiples of Two Numbers?

    Apr 15, 24 01:13 AM

    Common multiples of two or more given numbers are the numbers which can exactly be divided by each of the given numbers. Consider the following. (i) Multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24…

    Read More

  3. Least Common Multiple |Lowest Common Multiple|Smallest Common Multiple

    Apr 14, 24 03:06 PM

    Lowest Common Multiple
    The least common multiple (L.C.M.) of two or more numbers is the smallest number which can be exactly divided by each of the given number. The lowest common multiple or LCM of two or more numbers is t…

    Read More

  4. Worksheet on H.C.F. | Word Problems on H.C.F. | H.C.F. Worksheet | Ans

    Apr 14, 24 02:23 PM

    HCF Using Venn Diagram
    Practice the questions given in the worksheet on hcf (highest common factor) by factorization method, prime factorization method and division method. Find the common factors of the following numbers…

    Read More

  5. Common Factors | Find the Common Factor | Worksheet | Answer

    Apr 14, 24 02:01 PM

    Common Factors of 24 and 36
    Common factors of two or more numbers are a number which divides each of the given numbers exactly. For examples 1. Find the common factor of 6 and 8. Factor of 6 = 1, 2, 3 and 6. Factor

    Read More