The standard equation of a circle with center (h,k) and radius r is given by (x-h)² + (y-k)² = r². This formula is derived from the distance formula, where every point (x,y) on the circle is at a fixed distance r from the center (h,k).

The standard equation of a circle with center (h,k) and radius r is given by (x-h)² + (y-k)² = r². This formula is derived from the distance formula, where every point (x,y) on the circle is at a fixed distance r from the center (h,k).

Find the equation of the circle with centre (-2,7) and radius 5.

Answer: The equation of the circle is $(x+2)^2 + (y-7)^2 = 25$ .

Step-by-step solution

Step 1: Recall the Standard Equation of a Circle

The standard equation of a circle with center $(h,k)$ and radius $r$ is given by $(x-h)^2 + (y-k)^2 = r^2$ . This formula is derived from the distance formula, where every point $(x,y)$ on the circle is at a fixed distance $r$ from the center $(h,k)$ .

Step 2: Identify Given Values

From the problem statement, we are given that the center of the circle is $(-2{,}7)$ and the radius is $5$ . Therefore, we can identify the values for $h$ , $k$ , and $r$ .

Step 3: Substitute Values into the Equation

Now, substitute the identified values of $h=-2$ , $k=7$ , and $r=5$ into the standard equation of the circle. Be careful with the double negative for the $x$ -term.

Step 4: Simplify the Equation

Simplify the equation by resolving the double negative and calculating the square of the radius. This gives us the final equation of the circle.

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