Question

Which of the following is an equation of a circle in
the xy-plane with center (0, 4) and passes through (4/3,5)

Answer 1

Answer:

$x^2 + (y - 4)^2 = \dfrac{25}{9}$

Step-by-step explanation:

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

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

You have center (0, 4).

We get:

$(x - 0)^2 + (y - 4)^2 = r^2$

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

To find the radius, we use the distance formula to find the distance from the center of the circle to the given point on the circle.

$r = d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

$r = \sqrt{(\dfrac{4}{3} - 0)^2 + (5 - 4)^2}$

$r = \sqrt{(\dfrac{4}{3})^2 + 1^2}$

$r = \sqrt{\dfrac{16}{9} + \dfrac{9}{9}}$

$r = \sqrt{\dfrac{25}{9}}$

We need r^2 in the equation of the circle, so

$r^2 = \dfrac{25}{9}$

The equation of the circle is

$x^2 + (y - 4)^2 = \dfrac{25}{9}$