What is the equation of a circle with center #(2, -1)# that passes through the point #(3, 4)#?

1 Answer
smendyka
Jul 27, 2017

See a solution process below:

Explanation:

The equation for a circle is:

#(x - color(red)(a))^2 + (y - color(red)(b))^2 = color(blue)(r)^2#

Where #(color(red)(a), color(red)(b))# is the center of the circle and #color(blue)(2)# is the radius of the circle.

We know the center of the circle but we need to determine the radius of the circle.

The radius is the distance between the center of the circle and any point on the circle. We are given both the center of the circle and a point on the circle. We need to calculate the distance between these two points.

The formula for calculating the distance between two points is:

#d = sqrt((color(red)(x_2) - color(blue)(x_1))^2 + (color(red)(y_2) - color(blue)(y_1))^2)#

Substituting the values from the points in the problem gives:

#d = sqrt((color(red)(3) - color(blue)(2))^2 + (color(red)(4) - color(blue)(-1))^2)#

#d = sqrt((color(red)(3) - color(blue)(2))^2 + (color(red)(4) + color(blue)(1))^2)#

#d = sqrt(1^2 + 5^2)#

#d = sqrt(1 + 25)#

#d = sqrt(26)#

We can now substitute the values from the center point in the problem and the radius we calculated into the formula for the equation of a circle to give:

#(x - color(red)(2))^2 + (y - color(red)(-1))^2 = (color(blue)(sqrt(26)))^2#

#(x - color(red)(2))^2 + (y + color(red)(1))^2 = color(blue)(26)#

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