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. Conversion of Rupees and Paise | How to convert rupees into paise?

    Feb 12, 25 02:45 AM

    Conversion of Rupees and Paise
    We will discuss about the conversion of rupees and paise (i.e. from rupees into paise and from paise into rupees). How to convert rupees into paise? First we need to remove the point and then remove

    Read More

  2. Writing Money in Words and Figure Worksheet With Answers | All Grades

    Feb 12, 25 02:30 AM

    In writing money in words and figure worksheet we will get different types of questions on expressing money in words and figures, write the amount in short form (in figures) and write the amount in lo…

    Read More

  3. Writing Money in Words and Figure | Rules for Writing Money in Words

    Feb 12, 25 01:59 AM

    Rules for writing money in words and figure: 1. Abbreviation used for a rupee is Re. and for 1-rupee it is Re. 1 2. Rupees is written in short, as Rs., as 5-rupees is written as Rs. 5

    Read More

  4. Worksheet on Money | Conversion of Money from Rupees to Paisa

    Feb 11, 25 09:39 AM

    Amounts in Figures
    Practice the questions given in the worksheet on money. This sheet provides different types of questions where students need to express the amount of money in short form and long form

    Read More

  5. Worksheet on Measurement | Problems on Measurement | Homework |Answers

    Feb 10, 25 11:56 PM

    Measurement Worksheet
    In worksheet on measurement we will solve different types of questions on measurement of length, conversion of length, addition and subtraction of length, word problems on addition of length, word pro…

    Read More