Question

1. find the equation of the circle with center (0,0) and radius 3.

x + y = 3
x² + y² = 3
x² + y² = 9
x + y = 9

1. find the equation of the circle with center (-4,1) and radius 5.

(x - 4)² + (y + 1)² = 25
(x + 4)² + (y - 1)² = 5
(x + 4)² + (y - 1)² = 25

1. find the equation of the circle with center (1,3) and passes through (2,-4).

(x + 1)² + (y + 3)² = 50
(x - 1)² + (y - 3)² = 50
(x - 1)² + (y - 3)² = 8

Explanation:

Step1: Recall circle - equation formula

The standard form of the equation of a circle is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center of the circle and $r$ is the radius.

Step2: Solve problem 1

For a circle with center $(0,0)$ and radius $r = 3$, substituting $a = 0$, $b = 0$, and $r = 3$ into the formula, we get $(x-0)^2+(y - 0)^2=3^2$, which simplifies to $x^{2}+y^{2}=9$. But among the given options, the correct - form for a circle centered at the origin with radius $r$ is $x^{2}+y^{2}=r^{2}$, here $r = 3$, so the equation is $x^{2}+y^{2}=9$ (not in the options, but the correct form).

Step3: Solve problem 2

For a circle with center $(-4,1)$ and radius $r = 5$, substituting $a=-4$, $b = 1$, and $r = 5$ into the formula $(x - a)^2+(y - b)^2=r^2$, we have $(x+4)^2+(y - 1)^2=25$.

Step4: Solve problem 3

First, find the radius. The distance between the center $(1,3)$ and the point $(2,-4)$ is the radius $r$. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, where $(x_1,y_1)=(1,3)$ and $(x_2,y_2)=(2,-4)$. Then $r=\sqrt{(2 - 1)^2+(-4 - 3)^2}=\sqrt{1+( - 7)^2}=\sqrt{1 + 49}=\sqrt{50}$. The equation of the circle with center $(a = 1,b = 3)$ and radius $r=\sqrt{50}$ is $(x - 1)^2+(y - 3)^2=50$.

Answer:

1. No correct option among given ones (should be $x^{2}+y^{2}=9$)
2. $(x + 4)^2+(y - 1)^2=25$
3. $(x - 1)^2+(y - 3)^2=50$