Chapter 1.4, Problem 65AYU

(a)

To determine

To find: the center of the circle

Answer to Problem 65AYU

The center $\left(h,k\right)$ of a circle is the fixed point from which all the points lie on the circle keep a fixed distance (radius) $r$ .

Explanation of Solution

Explanations:

The standard equation of a circle with center $\left(h,k\right)$ and radius $4$ is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\mathrm{............}(1)$

The center $\left(h,k\right)$ of a circle is the fixed point from which all the points lie on the circle keep a fixed distance (radius) $r$ .

Conclusion:

The center $\left(h,k\right)$ of a circle is the fixed point from which all the points lie on the circle keep a fixed distance (radius) $r$ .

(b)

To determine

To find: the equation of the circle

Answer to Problem 65AYU

The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=5^2$ and the graph of the circle looks like:

  

Explanation of Solution

Given:The radius of the circle is $5$ .

Explanations:

The standard equation of a circle with center $\left(h,k\right)$ and radius $5$ is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\mathrm{............}(1)$

If the point $B$ is moved directly above the center $\left(h,k\right)$ such that the radius of the circle is $5$

The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=5^2$ and the graph of the circle looks like:

  

Conclusion:

Thus,The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=5^2$ and the graph of the circle looks like:

  

(c)

To determine

To find: the equation of the circle

Answer to Problem 65AYU

The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=4^2$ and the graph of the circle looks like:

  

Explanation of Solution

Given:The radius of the circle is $4$ .

Explanations:

The standard equation of a circle with center $\left(h,k\right)$ and radius $4$ is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\mathrm{............}(1)$

If the point $B$ is moved directly above the center $\left(h,k\right)$ such that the radius of the circle is $4$

The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=4^2$ and the graph of the circle looks like:

  

Conclusion:

Thus the equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=4^2$ and the graph of the circle looks like:

  

(d)

To determine

To find: the equation of the circle

Answer to Problem 65AYU

The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=3^2$ and the graph of the circle looks like:

  

Explanation of Solution

Given:The radius of the circle is $3$ .

Explanations:

The standard equation of a circle with center $\left(h,k\right)$ and radius $3$ is:

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2\mathrm{............}(1)$

If the point $B$ is moved directly above the center $\left(h,k\right)$ such that the radius of the circle is $3$

The equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=3^2$ and the graph of the circle looks like:

  

Conclusion:

Thus the equation of the circle becomes ${\left(x-h\right)}^2+{\left(y-k\right)}^2=3^2$ and the graph of the circle looks like:

  

(e)

To determine

To find: the coordinates of two points

Answer to Problem 65AYU

The coordinates of two points are $\left(5,-4\right)$ and $\left(4,-5\right)$

Explanation of Solution

Given:Radius of a Circle is 5

Explanations:

A center $\left(1,-1\right)$ obtained from $(a)$ and given radius is $5$ .

Now, the graph of a whose center is at $\left(1,-1\right)$ , radius is $5$ with two points $\left(5,-4\right)$ and $\left(4,-5\right)$ having integer coordinates and belonging to fourth quadrant is:

  

Also, solving the equation of the circle ${\left(x-1\right)}^2+{\left(y+1\right)}^2=5^2$ for $x$ and $y$ , there are two integer solutions belonging to the fourth quadrant are $\left(5,-4\right)$ and $\left(4,-5\right)$

Conclusion:

Thus, the coordinates of two points are $\left(5,-4\right)$ and $\left(4,-5\right)$

(f)

To determine

To find: three points with integer coordinates

Answer to Problem 65AYU

  $\left(5,2\right)$ [In I quadrant]

  $\left(-3,2\right)$ [In II quadrant]

  $\left(-3,-4\right)$ [In III quadrant]

Explanation of Solution

Given:center $\left(h,k\right)=\left(1,-1\right)$ and radius $r=5$ :

Explanations:

Suppose we have a center $\left(1,-1\right)$ obtained from $(a)$ and given radius is $5$ .

Now, the graph of a circle whose center is at $\left(1,-1\right)$ , radius is $5$ with two points $\left(5,-4\right)$ having integer coordinates and belonging to fourth quadrant is:

  

A circle is symmetry about the center, vertical line through the center of the circle and horizontal line through the center of the circle.

Now, for circle with center $\left(h,k\right)=\left(1,-1\right)$ and radius $r=5$ :

Taking $\left(5,-4\right)$ is a point in the IV quadrant,

  $\left(5,2\right)$ [in I quadrant]

  $\left(-3,2\right)$ [in II quadrant]

  $\left(-3,-4\right)$ [in III quadrant]

Conclusion:

Thus, the three points

  $\left(5,2\right)$ [In I quadrant]

  $\left(-3,2\right)$ [In II quadrant]

  $\left(-3,-4\right)$ [In III quadrant]