# Equation of a Circle when the Line Segment Joining Two Given Points is a Diameter

We will learn how to find the equation of the circle for which the line segment joining two given points is a diameter.

the equation of the circle drawn on the straight line joining two given points (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) as diameter is (x - x$$_{1}$$)(x - x$$_{2}$$)  + (y - y$$_{1}$$)(y - y$$_{2}$$) = 0

First Method:

Let P (x$$_{1}$$, y$$_{1}$$) and Q (x$$_{2}$$, y$$_{2}$$) are the two given given points on the circle. We have to find the equation of the circle for which the line segment PQ is a diameter.

Therefore, the mid-point of the line segment PQ is ($$\frac{x_{1} + x_{2}}{2}$$, $$\frac{y_{1} + y_{2}}{2}$$).

Now see that the mid-point of the line segment PQ is the centre of the required circle.

The radius of the required circle

= $$\frac{1}{2}$$PQ

= $$\frac{1}{2}$$$$\mathrm{\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}}$$

We know that the equation of a circle with centre at (h, k) and radius equal to a, is (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$.

Therefore, the equation of the required circle is

(x - $$\frac{x_{1} + x_{2}}{2}$$)$$^{2}$$ + (y - $$\frac{y_{1} + y_{2}}{2}$$)$$^{2}$$ = [$$\frac{1}{2}$$$$\mathrm{\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}}$$ ]$$^{2}$$

⇒ (2x - x$$_{1}$$ - x$$_{2}$$)$$^{2}$$ + (2y - y$$_{1}$$ - y$$_{2}$$)$$^{2}$$ = (x$$_{1}$$ - x$$_{2}$$)$$^{2}$$ + (y$$_{1}$$ - y$$_{2}$$)$$^{2}$$

⇒ (2x - x$$_{1}$$ - x$$_{2}$$)$$^{2}$$ - (x$$_{1}$$ - x$$_{2}$$)$$^{2}$$ + ( 2y - y$$_{1}$$ - y$$_{2}$$ )$$^{2}$$ - (y$$_{1}$$ - y$$_{2}$$)$$^{2}$$ = 0

⇒ (2x - x$$_{1}$$ - x$$_{2}$$ + x$$_{1}$$ - x$$_{2}$$)(2x - x$$_{1}$$ - x$$_{2}$$ - x$$_{1}$$ + x$$_{2}$$) + (2y - y$$_{1}$$ - y$$_{2}$$ + y$$_{1}$$ - y$$_{2}$$)(2y - y$$_{1}$$ - y$$_{2}$$ + y$$_{2}$$) = 0

⇒ (2x - 2x$$_{2}$$)(2x - 2x$$_{1}$$) + (2y - 2y$$_{2}$$)(2y - 2y$$_{1}$$) = 0

⇒ (x - x$$_{2}$$)(x - x$$_{1}$$) + (y - y$$_{2}$$)(y - y$$_{1}$$) = 0

⇒ (x - x$$_{1}$$)(x - x$$_{2}$$) + (y - y$$_{1}$$)(y - y$$_{2}$$) = 0.

Second Method:

equation of a circle when the co-ordinates of end points of a diameter are given

Let the two given points be P (x$$_{1}$$, y$$_{1}$$) and Q (x$$_{2}$$, y$$_{2}$$). We have to find the equation of the circle for which the line segment PQ is a diameter.

Let M (x, y) be any point on the required circle. Join PM and MQ.

m$$_{1}$$ = the slope of the straight line PM = $$\frac{y - y_{1}}{x - x_{1}}$$

m$$_{2}$$ = the slope of the straight line PQ = $$\frac{y - y_{2}}{x - x_{2}}$$.

Now, since the angle subtended at the point M in the semi-circle PMQ is a right angle.

Now, PQ is a diameter of the required circle.

Therefore, ∠PMQ = 1 rt. angle i.e., PM is perpendicular to QM

Therefore, $$\frac{y - y_{1}}{x - x_{1}}$$ × $$\frac{y - y_{2}}{x - x_{2}}$$ = -1

(y - y$$_{1}$$)(y - y$$_{2}$$) = - (x - x$$_{1}$$)(x - x$$_{2}$$

(x - x$$_{1}$$)(x - x$$_{2}$$) + (y - y$$_{1}$$)(y - y$$_{2}$$) = 0.

This is the required equation of the circle having (x$$_{1}$$, y$$_{1}$$) and (x$$_{2}$$, y$$_{2}$$) as the coordinates of the end points of a diameter.

Note: If the coordinates of the end points of a diameter of a circle given, we can also find the equation of the circle by finding the coordinates of the centre and radius. The centre is the mid-point of the diameter and radius is half of the length of the diameter.

The Circle

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.

## Recent Articles

1. ### Fraction as a Part of Collection | Pictures of Fraction | Fractional

Feb 24, 24 04:33 PM

How to find fraction as a part of collection? Let there be 14 rectangles forming a box or rectangle. Thus, it can be said that there is a collection of 14 rectangles, 2 rectangles in each row. If it i…

2. ### Fraction of a Whole Numbers | Fractional Number |Examples with Picture

Feb 24, 24 04:11 PM

Fraction of a whole numbers are explained here with 4 following examples. There are three shapes: (a) circle-shape (b) rectangle-shape and (c) square-shape. Each one is divided into 4 equal parts. One…

3. ### Identification of the Parts of a Fraction | Fractional Numbers | Parts

Feb 24, 24 04:10 PM

We will discuss here about the identification of the parts of a fraction. We know fraction means part of something. Fraction tells us, into how many parts a whole has been

4. ### Numerator and Denominator of a Fraction | Numerator of the Fraction

Feb 24, 24 04:09 PM

What are the numerator and denominator of a fraction? We have already learnt that a fraction is written with two numbers arranged one over the other and separated by a line.