A circle centered at point O (-4; 0) passes through point K (-1; 4) a) write down the equation

A circle centered at point O (-4; 0) passes through point K (-1; 4) a) write down the equation of this circle. b) find the points of the circle that have an ordinate equal to 3.

a) Since the center of this circle lies at point O with coordinates (-4; 0) and this circle passes through point K with coordinates (-1; 4), the radius of this circle is equal to the length of the segment OK.

We find the length of this segment using the formula for the distance between two points on the coordinate plane:

| OK | = √ ((- 4 – (-1)) ^ 2 + (4 – 0) ^ 2) = √ ((- 4 + 1) ^ 2 + 4 ^ 2) = √ ((- 3) ^ 2 + 4 ^ 2) = √ (9 + 65) = √25 = 5.

We write down the equation of the circle:

(x – (-4)) ^ 2 + (y – 0) ^ 2 = 5 ^ 2.

Simplifying this ratio, we get:

(x + 4) ^ 2 + y ^ 2 = 25.

Answer: The equation of the circle is (x + 4) ^ 2 + y ^ 2 = 25.

b) Substituting the value y = 4 into this equation, we find the points of the circle that have an ordinate equal to 3:

(x + 4) ^ 2 + 3 ^ 2 = 25;

(x + 4) ^ 2 + 9 = 25;

(x + 4) ^ 2 + 9 – 25 = 0;

(x + 4) ^ 2 – 16 = 0;

(x + 4) ^ 2 – 4 ^ 2 = 0;

(x + 4 – 4) * (x + 4 + 4) = 0;

x * (x + 8) = 0:

x1 = 0;

x2 = -8.

Therefore, the required points have coordinates (0; 3) and (-8; 3).

Answer: the required points have coordinates (0; 3) and (-8; 3).