Q. 12 - Geometry

Question

12, Find an equation of the circle that has the given center and radius

a. Center (2, 5); radius 3.

b. Center (-2, 0); radius 5.

c. Center (-2, 3); radius 10. 

d. Center (j, 4); radius n.

Step-by-Step Solution

Answer

1. The equation of the circle is ${\left(x-2\right)}^2+{\left(y-5\right)}^2=9$.
2. The equation of the circle is${\left(x+2\right)}^2+y^2=25$ .
3. The equation of the circle is ${\left(x+2\right)}^2+{\left(y-3\right)}^2=100$.
4. The equation of the circle is${\left(x-j\right)}^2+{\left(y-k\right)}^2=n^2$ .

a. Step-1 – Given

The given center = (2, 5) and radius = 3.

Step-2 – To determine

We have to find the equation of the circle.

Step-3 – Calculation

We know that the standard form of the equation of a circle with center (a, b) and radius r is${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ .

From the given equation, the center (a, b) = (2, 5) and radius = r = 3.

Plug them in the standard form of the circle.

So, the equation of the circle is${\left(x-2\right)}^2+{\left(y-5\right)}^2={\left(3\right)}^2$ .

Or,${\left(x-2\right)}^2+{\left(y-5\right)}^2=9$ .

b. Step-1 – Given

The given center = (-2, 0) and radius = 5.

Step-2 – To determine

We have to find the equation of the circle.

Step-3 – Calculation

We know that the standard form of the equation of a circle with center (a, b) and radius r is${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ .

From the given equation, the center (a, b) = (-2, 0) and radius = r = 5.

Plug them in the standard form of the circle.

So, the equation of the circle is${\left(x+2\right)}^2+y^2={\left(5\right)}^2$ .

Or,${\left(x+2\right)}^2+y^2=25$ .

c. Step-1 – Given

The given center = (-2, 3) and radius = 10.

Step-2 – To determine

We have to find the equation of the circle.

Step-3 – Calculation

We know that the standard form of the equation of a circle with center (a, b) and radius r is${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ .

From the given equation, the center (a, b) = (-2, 3) and radius = r = 10.

Plug them in the standard form of the circle.

So, the equation of the circle is${\left(x+2\right)}^2+{\left(y-3\right)}^2={\left(10\right)}^2$ .

Or,${\left(x+2\right)}^2+{\left(y-3\right)}^2=100$ .

d. Step-1 – Given

The given center = (j, k) and radius = n.

Step-2 – To determine

We have to find the equation of the circle.

Step-3 – Calculation

We know that the standard form of the equation of a circle with center (a, b) and radius r is ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$.

From the given equation, the center (a, b) = (j, k) and radius = r = n.

Plug them in the standard form of the circle.

So, the equation of the circle is ${\left(x-j\right)}^2+{\left(y-k\right)}^2={\left(n\right)}^2$.

Or, ${\left(x-j\right)}^2+{\left(y-k\right)}^2=n^2$.