Two circles centered at (2,3) and (4,7) intersect each other. If their radii are equal, then the equation of their common chord is ?

  1. 2x + y = 15
  2. X – y = 9
  3. X + y = 8
  4. X+2y=13
  5. -2x + y = 5

1 Answer
CW
Feb 17, 2018

x+2y=13

Explanation:

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As shown in the figure, A(2,3) is the center of Circle 1 and B(4,7) the center of Circle 2, PQ is the common chord.
Given that the circles' radii are equal,
C(x_1,y_1) is the midpoint of AB. It is also the midpoint of the common chord PQ.
=> coordinates of C(x_1,y_1)=((4+2)/2, (7+3)/2)=(3,5)
We know that the line joining the center of a circle and the midpoint of a chord is perpendicular to the chord,
=> PQ is perpendicular to AB
Let m_(AB) and m_(PQ) be the slope of AB and PQ, respectively,
=> m_(AB)=(7-3)/(4-2)=4/2=2
As the product of the slopes of two perpendicular lines is -1,
=> m_(AB)*m_(PQ)=-1
=> m_(PQ)=-1/2
Equation of a line that passes through C(x_1,y_1) with a slope m is given by :
y-y_1=m(x-x_1)
=> y-5=-1/2(x-3)
=> 2y-10=-x+3
=> x+2y=13

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