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.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Addition and Subtraction of Fractions | Solved Examples | Worksheet

    Jul 18, 24 03:08 PM

    Addition and subtraction of fractions are discussed here with examples. To add or subtract two or more fractions, proceed as under: (i) Convert the mixed fractions (if any.) or natural numbers

    Read More

  2. Worksheet on Simplification | Simplify Expressions | BODMAS Questions

    Jul 18, 24 01:19 AM

    In worksheet on simplification, the questions are based in order to simplify expressions involving more than one bracket by using the steps of removal of brackets. This exercise sheet

    Read More

  3. Fractions in Descending Order |Arranging Fractions an Descending Order

    Jul 18, 24 01:15 AM

    We will discuss here how to arrange the fractions in descending order. Solved examples for arranging in descending order: 1. Arrange the following fractions 5/6, 7/10, 11/20 in descending order. First…

    Read More

  4. Fractions in Ascending Order | Arranging Fractions | Worksheet |Answer

    Jul 18, 24 01:02 AM

    Comparison Fractions
    We will discuss here how to arrange the fractions in ascending order. Solved examples for arranging in ascending order: 1. Arrange the following fractions 5/6, 8/9, 2/3 in ascending order. First we fi…

    Read More

  5. Worksheet on Comparison of Like Fractions | Greater & Smaller Fraction

    Jul 18, 24 12:45 AM

    Worksheet on Comparison of Like Fractions
    In worksheet on comparison of like fractions, all grade students can practice the questions on comparison of like fractions. This exercise sheet on comparison of like fractions can be practiced

    Read More