Question

Determine the equation of the circle with center left bracket, minus, 2, comma, minus, 3, right bracket(−2,−3) containing the point left bracket, 4, comma, 5, right bracket(4,5).

Answer 1

Explanation:

The equation of a circle in the Cartesian coordinate system with center \((h, k)\) and radius \(r\) is given by the formula:

\[(x - h)^2 + (y - k)^2 = r^2\]

In your case, the center of the circle is \((-2, -3)\), and it contains the point \((4, 5)\). To find the radius (\(r\)), you can use the distance formula between two points:

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

Let's substitute the given values:

\[r = \sqrt{(4 - (-2))^2 + (5 - (-3))^2}\]

\[r = \sqrt{6^2 + 8^2}\]

\[r = \sqrt{36 + 64}\]

\[r = \sqrt{100}\]

\[r = 10\]

Now, substitute the center and radius into the circle equation:

\[(x - (-2))^2 + (y - (-3))^2 = 10^2\]

\[(x + 2)^2 + (y + 3)^2 = 100\]

So, the equation of the circle is \((x + 2)^2 + (y + 3)^2 = 100\).