How do you find the equation of a circle with center on the line x + 2y= 4 passes through the points A(6,9) and B(12,-9)?

Question

Madelyn Anderson · Accepted Answer

The equation of circle is #(x - 6)^2 + (y +1)^2 = 10^2# Let #(x,y)# be the centre of the circle which passes through points #A(6,9) and B(12,-9) # and #r# be the radius of the circle. Distance formula is #D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2# #:.r^2=(x-6)^2+(y-9)^2 and r^2=(x-12)^2+(y+9)^2 # #:. (x-6)^2+(y-9)^2 =(x-12)^2+(y+9)^2# or #cancel(x^2)-12x+36+cancel(y^2)-18y+cancel81=cancel(x^2)-24x+144+cancel(y^2)+18y+cancel81# or #24x-12x-18y-18y=144-36 or 12x-36y=108# or #x-3y=9 (1) # The centre #(x,y)# also satisfy the equation #x+2y=4 (2)# . Solving the equation (1) and equation (2) we get #x=6 , y=-1# Hence center is at #(6,-1)# and radius #:. r^2=(6-6)^2+(-1-9)^2=100 :. r=10# The equation of circle is #(x – h)^2 + (y – k)^2 = r^2# with the center being at the point #(h=6, k=-1)# and the radius being #r#. Equation of circle is #:.(x - 6)^2 + (y +1)^2 = 10^2# [Ans]

Ariana Alvarez · Answer

undefined To find the equation of a circle with its center on the line $x + 2y = 4$ that passes through the points A(6,9) and B(12,-9):

1. Find the midpoint of the line segment connecting points A and B. This midpoint will be the center of the circle.

2. Use the midpoint coordinates as the center of the circle.

3. Calculate the radius of the circle by finding the distance between either point A or B and the center (midpoint) using the distance formula.

4. Substitute the center coordinates and radius into the standard form equation of a circle $ (x - h)^2 + (y - k)^2 = r^2 $, where (h, k) are the coordinates of the center and r is the radius.

5. Simplify the equation to find the final equation of the circle.