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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