Chapter 1.4, Problem 22AYU

To determine

To find:

1. The centre $\left(h,k\right)$ and radius of the circle $x²+\left(y-1\right)²=1$

2. The graph of the circle

3. Intercepts of the circle, if any

Answer to Problem 22AYU

$\text{Centre }\left(0,1\right);\text{ radius }=1;$

$x\text{-intercept }=0;y\text{-intercept }=2$

Explanation of Solution

Given:

Equation of the circle $x²+\left(y-1\right)²=1$

Calculation:-

Let us write the equation of the circle in standard form $\left(x-0\right)²+(y-1)²=\left(1\right)²$.

Compare this equation with the equation $\left(x-h\right)²+(y-k)²=\left(r\right)²$.

The comparison yields the information about the circle. We see that $h=0,k=1$ and $r=1$.

The circle has centre $\left(0,1\right)$ and a radius 1 units. To graph the circle, first plot the centre $\left(0,1\right)$. Since the radius is 1 units, locate four points on the circle by plotting 1 units to the left, to the right, up and down from the centre. These four points can be used to sketch the graph.

To find the $x\text{-intercepts}$, if any, let $y=0$ and solve for $x$.

$x²+\left(0-1\right)²=1$

$x²+1=1$

$x²+1-1=1-1$   Subtract 1 from both sides

$x²=0$

$x=0$

$x\text{-intercepts}$ is 0.

To find the $y\text{-intercepts}$, if any, let $x=0$ and solve for $y$.

$\left(0\right)²+\left(y-1\right)²=1$

$y-1=\pm 1$

$\text{If }y-=1$

$y-1+1=1+1$

$y=2$

$\text{If }y-1=-1$

$y-1+1=\mathrm{-1}+1$

$y=0$

$y\text{-intercepts}$ is 2.