Identify the equation of the circle y that passes through (2,6) and has center (3,4).

(x − 3)² + (y − 4)² = 5

Step-by-step explanation:

The equation of a circle is:

(x − h)² + (y − k)² = r²

where (h, k) is the center and r is the radius.

First use the distance formula to find the radius:

d² = (x₂ − x₁)² + (y₂ − y₁)²

r² = (2 − 3)² + (6 − 4)²

r² = 1 + 4

r² = 5

Given that (h, k) = (3, 4):

(x − 3)² + (y − 4)² = 5

Step-by-step explanation:

Inserting the coordinates of the center (3, 4) into the standard equation of a circle with center at (h, k) and radius r, we get:

(x - 3)^2 + (y - 4)^2 = r^2

Next, we substitute 2 for x, 6 for y and solve the resulting equation for r^2:

(2 - 3)^2 + (6 - 4)^2 = r^2, or

      1      +       4       =  r^2.

Thus, the radius is √5.  Subbing this result into the equation found above, (x - 3)^2 + (y - 4)^2 = r^2, we get:

(x - 3)^2 + (y - 4)^2 = (√5)^2 = 5, which matches the last of the four possible answer choices.