Question

Write the equation of the circle with center (2, 1) and (5, 2) a point on the circle

Answer 1

The equation of a circle centered at (h, k) and with radius r is

$(x - h)^2 + (y - k)^2 = r^2$

In this case, the center is point (2, 1), so h = 2, and k = 1.

We also need the radius. We are not given the radius, but we have the center and a point on the circle. The distance between the center and any point on the circle is the radius. We can use the distance formula to find the distance between points (2, 1) and (5, 2).

$d = \sqrt{(x_2 - x_1)^2 + (x_2 - x_1)^2}$

$d = \sqrt{(5 - 2)^2 + (2 - 1)^2}$

$d = \sqrt{3^2 + 1^2}$

$d = \sqrt{9 + 1}$

$d = \sqrt{10}$

The radius is the distance between the points, so $r = \sqrt{10}$

Now we have all the information we need, so we can write the equation of the circle.

$(x - 2)^2 + (y - 1)^2 = \sqrt{10^2$

$(x - 2)^2 + (y - 1)^2 = 10$

Answer:     $(x - 2)^2 + (y - 1)^2 = 10$