Subscribe to our YouTube channel for the latest videos, updates, and tips.


Circle Passes through the Origin

We will learn how to form the equation of a circle passes through the origin.

The equation of a circle with centre at (h, k) and radius equal to a, is (x - h)\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\).

When the centre of the circle coincides with the origin i.e., a\(^{2}\) = h\(^{2}\) + k\(^{2}\)

Let O be the origin and C(h, k) be the centre of the circle. Draw CM perpendicular to OX.

In triangle OCM, OC\(^{2}\) = OM\(^{2}\) + CM\(^{2}\)

i.e., a\(^{2}\) = h\(^{2}\) + k\(^{2}\).


Therefore, the equation of the circle (x - h)\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\) becomes

(x - h)\(^{2}\) + (y - k)\(^{2}\) = h\(^{2}\) + k\(^{2}\)

⇒ x\(^{2}\) + y\(^{2}\) - 2hx – 2ky = 0

The equation of a circle passing through the origin is

x\(^{2}\) + y\(^{2}\) + 2gx + 2fy = 0 ……………. (1)

or, (x - h)\(^{2}\) + (y - k)\(^{2}\) = h\(^{2}\) + k\(^{2}\) …………………………. (2)

 We clearly see that the equations (1) and (2) are satisfied by (0, 0).

 

Solved examples on the central form of the equation of a circle passes through the origin:

1. Find the equation of a circle whose centre is (2, 3) and passes through the origin.

Solution:

The equation of a circle with centre at (h, k) and passes through the origin is

(x - h)\(^{2}\) + (y - k)\(^{2}\) = h\(^{2}\) + k\(^{2}\)

Therefore, the required equation of the circle is (x - 2)\(^{2}\) + (y - 3)\(^{2}\) = 2\(^{2}\) + 3\(^{2}\)

⇒ x\(^{2}\) - 4x + 4 + y\(^{2}\) – 6y + 9 = 4 + 9

⇒ x\(^{2}\) + y\(^{2}\) - 4x – 6y = 0.


2. Find the equation of a circle whose centre is (-5, 4) and passes through the origin.

Solution:

The equation of a circle with centre at (h, k) and passes through the origin is

(x - h)\(^{2}\) + (y - k)\(^{2}\) = h\(^{2}\) + k\(^{2}\)

Therefore, the required equation of the circle is (x + 5)\(^{2}\) + (y - 4)\(^{2}\) = (-5)\(^{2}\) + 4\(^{2}\)

⇒ x\(^{2}\) + 10x + 25 + y\(^{2}\) – 8y + 16 = 25 + 16

⇒ x\(^{2}\)+ y\(^{2}\) + 10x – 8y = 0.

 The Circle





11 and 12 Grade Math 

From Circle Passes through the Origin 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. Calculating Profit Percent and Loss Percent | Profit and Loss Formulas

    Jun 12, 25 12:48 PM

    In calculating profit percent and loss percent we will learn about the basic concepts of profit and loss. We will recall facts and formula while calculating profit percent and loss percent. Now we wil

    Read More

  2. Word Problems on Profit and Loss Worksheet |Cost Price |Selling Price

    Jun 11, 25 04:26 PM

    Word Problems on Profit and Loss Worksheet
    In word problems on profit and loss worksheet you will get different types of problems on cost price and selling price, profit and loss, calculating profit o loss, calculating selling price and cost p…

    Read More

  3. Round off to Nearest 1000 |Rounding Numbers to Nearest Thousand| Rules

    Jun 11, 25 03:12 PM

    Round off to Nearest 1000
    Round off to nearest 1000 is discussed here. While rounding off to the nearest 1000, if the digit in the hundreds place is between 0 – 4 i.e., < 5, then the hundreds place is replaced by ‘0’. If the d…

    Read More

  4. Round off to Nearest 100 | Rounding Numbers To Nearest Hundred | Rules

    Jun 11, 25 03:13 AM

    Round off to Nearest 100
    While rounding off to the nearest hundred, if the digit in the tens place is between 0 – 4 i.e. < 5, then the tens place is replaced by ‘0’. If the digit in the units place is equal to or >5, then the…

    Read More

  5. Round off to Nearest 10 |How To Round off to Nearest 10?|Rounding Rule

    Jun 10, 25 05:36 PM

    Rounding to the Nearest 10
    Round off to nearest 10 is discussed here. Rounding can be done for every place-value of number. To round off a number to the nearest tens, we round off to the nearest multiple of ten. A large number…

    Read More