A circle of center (-3 , -2) passes through the points (0 , -6) and (a , 0). Find a.

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From the given information,

radius of circle

= sqrt{ (-3-0)^2 + [-2-(-6)]^2 }

= sqrt{ 9 + 16 }

= sqrt(25)

= 5

The equation of the circle would be:

(x+3)^2 + (y+2)^2 = 5^2

At coordinate (a,0), x=a and y=0.

Substituting y=0 into the equation of circle,

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From the given information,

radius of circle

= sqrt{ (-3-0)^2 + [-2-(-6)]^2 }

= sqrt{ 9 + 16 }

= sqrt(25)

= 5

 

The equation of the circle would be:

(x+3)^2 + (y+2)^2 = 5^2

 

At coordinate (a,0), x=a and y=0.

Substituting y=0 into the equation of circle,

(x+3)^2 + (0+2)^2 = 5^2

x^2 + 6x + 9 + 4 = 25

x^2 + 6x - 12 = 0

a1 = { -6 + sqrt[ 36 - 4*(-12) ] }/2 = 1.5826 (correct to 4 d.p.)

a2 = { -6 - sqrt[ 36 - 4*(-12) ] }/2 = -7.5826 (correct to 4 d.p.)

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A circle of center (-3 , -2) passes through the points (0 , -6) and (a , 0). Find a

First we will write the standard form for the circle:

 ( x- x1)^2 + (y-y^2) = r^2  such that:

(x1, y1) is the center and r is the radius:

Given the center C( -3,-2) then we will substitute:

==> (x+3)^2 + (y+2)^2 = r^2

Now we need to calculate r:

Give that the points A(0, -6) and B(a,0) pass throughthe circle:

We know that the distance between any point on the circle and the center equals the radius:

==> r = l AC l = l BCl

Let us calculate:

l AC l = sqrt( 0+3)^2 + ( -6+2)^2

            = sqrt(9 + 16) = sqrt25 = 5

Then the radius r = 5

l BC l = sqrt( a+ 3)^2 + ( 0 + 2)^2 = 5

==> sqrt( a^2 + 6a + 9 + 4) = 5

==> sqrt(a^2 + 6a + 13) = 5

Square both sides:

==> a^2 + 6a + 13 = 25

==> a^2 + 6a - 12 = 0

==> a1 = ( -6 + sqrt(84) /2 = ( -6 + 2sqrt21)/2 = -3 + sqrt21

==> a2= -3 - sqrt21

Approved by eNotes Editorial Team