Question

Which of the following is the equation of the circle with center C and a point P on the circle, as shown in the graph?

segment CP with endpoints C at negative 1 comma 1 and P at negative 5 comma negative 2 with a length of r equals 5
a. (x + 1)2 + (y – 1)2 = 25
b. (x + 5)2 + (y + 2)2 = 25
c. (x – 1)2 + (y + 1)2 = 5
d. (x – 5)2 + (y – 2)2 = 5

Answer 1

The correct equation for the circle with center C and a point P on the circle is
`\((x - 1)^2 + (y + 1)^2 = 25\)`, and the answer is option (c).

The equation of a circle with center (h, k) and radius r is given by the formula
`\((x - h)^2 + (y - k)^2 = r^2\)`. In this case, the center C is at (-1, 1) and the radius is 5.

The correct equation is obtained as follows:

`\[(x + 1)^2 + (y - 1)^2 = 5^2\]`

Expanding and simplifying:

`\[x^2 + 2x + 1 + y^2 - 2y + 1 = 25\]`

Combining like terms:

`\[x^2 + y^2 + 2x - 2y - 23 = 0\]`

Rearranging:

`\[x^2 + 2x + y^2 - 2y = 23\]`

Comparing with the given options:

a. Incorrect

b. Incorrect

c. Incorrect

d. Correct

Therefore, the correct equation for the circle with center C and a point P on the circle is
`\((x - 1)^2 + (y + 1)^2 = 25\)`, and the answer is option (c).