Equation of the Common Chord of Two Circles

We will learn how to find the equation of the common chord of two circles.

Let us assume that the equations of the two given intersecting circles be x\(^{2}\) + y\(^{2}\) + 2g\(_{1}\)x + 2f\(_{1}\)y + c\(_{1}\) = 0 ……………..(i) and x\(^{2}\) + y\(^{2}\) + 2g\(_{2}\)x + 2f\(_{2}\)y + c\(_{2}\) = 0 ……………..(ii), intersect at P (x\(_{1}\), y\(_{1}\)) and Q (x\(_{2}\), y\(_{2}\)).

Now we need to find the equation of the common chord PQ of the given circles.

Now we observe from the above figure that the point P (x\(_{1}\), y\(_{1}\)) lies on both the given equations. 

Therefore, we get,

x\(_{1}\)\(^{2}\) + y\(_{1}\)\(^{2}\) + 2g\(_{1}\)x\(_{1}\) + 2f\(_{1}\)y\(_{1}\) + c\(_{1}\) = 0 ……………..(iii)    


x\(_{1}\)\(^{2}\) + y\(_{1}\)\(^{2}\) + 2g\(_{2}\)x\(_{1}\) + 2f\(_{2}\)y\(_{1}\) + c\(_{2}\) = 0 ……………..(iv)

Now subtracting the equation (4) from equation (3) we get,

2(g\(_{1}\) -  g\(_{2}\))x\(_{1}\) + 2 (f\(_{1}\) - f\(_{2}\))y\(_{1}\) + C\(_{1}\) - C\(_{2}\) = 0 ……………..(v)

Again, we observe from the above figure that the point Q (x2, y2) lies on both the given equations. Therefore, we get,


x\(_{2}\)\(^{2}\) + y\(_{2}\)\(^{2}\) + 2g\(_{1}\)x\(_{2}\) + 2f\(_{1}\)y\(_{2}\) + c\(_{1}\) = 0 ……………..(vi)


x\(_{2}\)\(^{2}\) + y\(_{2}\)\(^{2}\) + 2g\(_{2}\)x\(_{2}\) + 2f\(_{2}\)y\(_{2}\) + c\(_{2}\) = 0 ……………..(vii)

Now subtracting the equation (b) from equation (a) we get,

2(g\(_{1}\) -  g\(_{2}\))x\(_{2}\) + 2 (f\(_{1}\) - f\(_{2}\))y\(_{2}\) + C\(_{1}\) - C\(_{2}\) = 0 ……………..(viii)

From conditions (v) and (viii) it is evident that the points P (x\(_{1}\), y\(_{1}\)) and Q (x\(_{2}\), y\(_{2}\)) lie on 2(g\(_{1}\) -  g\(_{2}\))x + 2 (f\(_{1}\) - f\(_{2}\))y + C\(_{1}\) - C\(_{2}\) = 0, which is a linear equation in x and y.

It represents the equation of the common chord PQ of the given two intersecting circles.

 

Note: While finding the equation of the common chord of two given intersecting circles first we need to express each equation to its general form i.e., x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 then subtract one equation of the circle from the other equation of the circle.


Solve example to find the equation of the common chord of two given circles:

1. Determine the equation of the common chord of the two intersecting circles x\(^{2}\) + y\(^{2}\) - 4x - 2y - 31 = 0 and 2x\(^{2}\) + 2y\(^{2}\) - 6x + 8y - 35 = 0 and prove that the common chord is perpendicular to the line joining the centers of the two circles.

Solution:

The given two intersecting circles are

x\(^{2}\) + y\(^{2}\) - 4x - 2y - 31 = 0 ……………..(i) and

2x\(^{2}\) + 2y\(^{2}\) - 6x + 8y - 35 = 0              

⇒ x\(^{2}\) + y\(^{2}\) - 3x + 4y - \(\frac{35}{2}\) ……………..(ii)

Now, to find the equation of the common chord of two intersecting circles we will subtract the equation (ii) from the equation (i).

Therefore, the equation of the common chord is

x\(^{2}\) + y\(^{2}\) - 4x - 2y - 31 - (x\(^{2}\) + y\(^{2}\) - 3x + 4y - \(\frac{35}{2}\)) = 0    

⇒ - x - 6y - \(\frac{27}{2}\) = 0            

2x + 12y + 27  = 0, which is the required equation.

The slope of the common chord 2x + 12y + 27 = 0 is (m\(_{1}\)) = -\(\frac{1}{6}\).

Centre of the circle x\(^{2}\) + y\(^{2}\) - 4x - 2y - 31 = 0 is (2, 1).

Centre of the circle 2x\(^{2}\) + 2y\(^{2}\) - 6x + 8y - 35 = 0 is (\(\frac{3}{2}\), -2).

The slope of the line joining the centres of the circles (1) and (2) is (m\(_{2}\)) = \(\frac{-2 - 1}{\frac{3}{2} - 2}\) = 6

Now m\(_{1}\) ∙ m\(_{2}\) = -\(\frac{1}{6}\) ∙ 6 = - 1

Therefore, we see that the slope of the common chord and slope of the line joining the centres of the circles (1) and (2) are negative reciprocals of each other i.e., m\(_{1}\) = -\(\frac{1}{m_{2}}\) i.e., m\(_{1}\) ∙ m\(_{2}\) = -1.

Therefore, the common chord of the given circles is perpendicular to the line joining the centers of the two circles.              Proved

 The Circle




11 and 12 Grade Math 

From Equation of the Common Chord of Two Circles 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. Patterns in Numbers | Patterns in Maths |Math Patterns|Series Patterns

    Dec 13, 24 08:43 AM

    Complete the Series Patterns
    We see so many patterns around us in our daily life. We know that a pattern is an arrangement of objects, colors, or numbers placed in a certain order. Some patterns neither grow nor reduce but only r…

    Read More

  2. Patterns in Math | Missing Number | Counting Numbers | Worksheets

    Dec 13, 24 12:31 AM

    Finding patterns in math is very important to understand the sequence in the series. We need to find the exact missing number that from the group of numbers. The counting numbers may be counting

    Read More

  3. Concept of Pattern | Similar Patterns in Mathematics | Similar Pattern

    Dec 12, 24 11:22 PM

    Patterns in Necklace
    Concept of pattern will help us to learn the basic number patterns and table patterns. Animals such as all cows, all lions, all dogs and all other animals have dissimilar features. All mangoes have si…

    Read More

  4. 2nd Grade Geometry Worksheet | Plane and Solid Shapes | Point | Line

    Dec 12, 24 10:31 PM

    Curved Line and Straight Line
    2nd grade geometry worksheet

    Read More

  5. Types of Lines |Straight Lines|Curved Lines|Horizontal Lines| Vertical

    Dec 09, 24 10:39 PM

    Types of Lines
    What are the different types of lines? There are two different kinds of lines. (i) Straight line and (ii) Curved line. There are three different types of straight lines. (i) Horizontal lines, (ii) Ver…

    Read More