Q. 11 - Geometry

Question

11. Find the center and the radius of each circle.

a. ${(x-2)}^2+y^2=1$

b. ${(x+2)}^2+{(y-8)}^2=16$

c. $x^2+{(y+5)}^2=112$

d. ${(x+3)}^2+{(y+7)}^2=14$

Step-by-Step Solution

Answer

1. The center is (2, 0) and the radius is 1.
2. The center is (-2, 8) and the radius is 4.
3. The center is (0, -5) and the radius is$\sqrt{112}$ .
4. The center is (-3, -7) and the radius is$\sqrt{14}$ .

a. Step-1 – Given

The given equation is ${\left(x-2\right)}^2+y^2=1$

Step-2 – To determine

We have to find the center and radius of a circle.

Step-3 – Calculation

Compare the given circle:${\left(x-2\right)}^2+y^2=1$ with the standard form of a circle:${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ .

Comparing we get: a = 2, b = 0 and r = 1.

So, the center is (2, 0) and the radius is 1.

b. Step-1 – Given

The given equation is ${\left(x+2\right)}^2+{\left(y-8\right)}^2=16$

Step-2 – To determine

We have to find the center and radius of a circle.

Step-3 – Calculation

Compare the given circle:${\left(x+2\right)}^2+{\left(y-8\right)}^2=16$ with the standard form of a circle: ${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$.

Comparing we get: a = -2, b = 8 and r = 4.

So, the center is (-2, 8) and the radius is 4.

c. Step-1 – Given

The given equation is$x^2+{\left(y+5\right)}^2=112$ .

Step-2 – To determine

We have to find the center and radius of a circle.

Step-3 – Calculation

Compare the given circle:$x^2+{\left(y+5\right)}^2=112$ with the standard form of a circle:${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ .

Comparing we get: a = 0, b = -5 and r =$\sqrt{112}$ .

So, the center is (0, -5) and the radius is$\sqrt{112}$ .

d. Step-1 – Given

The given equation is ${\left(x+3\right)}^2+{\left(y+7\right)}^2=14$.

Step-2 – To determine

We have to find the center and radius of a circle.

Step-3 – Calculation

Compare the given circle:${\left(x+3\right)}^2+{\left(y+7\right)}^2=14$ with the standard form of a circle:${\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2$ .

Comparing we get: a = -3, b = -7 and r =$\sqrt{14}$ .

So, the center is (-3, -7) and the radius is$\sqrt{14}$ .