Chapter 1, Problem 99RE

a.

To determine

Plot the points on coordinate plane.

Answer to Problem 99RE

  

Explanation of Solution

Given information:

The given points are, $P\left(2,0\right),Q\left(-5,12\right)$ .

Calculation:

Consider the given points,

  $P\left(2,0\right),Q\left(-5,12\right)$

Now plot the points on coordinate plane.

  

Hence, the points are plotted.

b.

To determine

Find the distance between the points.

Answer to Problem 99RE

  $d=\sqrt{193}$ .

Explanation of Solution

Given information:

The given points are, $P\left(2,0\right),Q\left(-5,12\right)$ .

Calculation:

Consider the given points,

  $P\left(2,0\right),Q\left(-5,12\right)$

Now apply distance formula,

  $\begin{array}{l}d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ d=\sqrt{{\left(-5-2\right)}^2+{\left(12-0\right)}^2}\\ =\sqrt{{\left(-7\right)}^2+{\left(12\right)}^2}\\ =\sqrt{49+144}\\ d=\sqrt{193}\end{array}$

Hence, the distance between the points is $d=\sqrt{193}$ .

c.

To determine

Find the midpoint of line segment PQ.

Answer to Problem 99RE

  $-3/2,6$ .

Explanation of Solution

Given information:

The given points are, $P\left(2,0\right),Q\left(-5,12\right)$ .

Calculation:

Consider the given points,

  $P\left(2,0\right),Q\left(-5,12\right)$

Now apply midpoint formula for points $\left(x_1,y_1\right),\left(x_2,y_2\right)$ ,

  $\begin{array}{l}x=\frac{x_1+x_2}{2},y=\frac{y_1+y_2}{2}\\ x=\frac{2-5}{2},y=\frac{0+12}{2}\\ x=-3/2,y=6\end{array}$

Hence, midpoints are $-3/2,6$ .

d.

To determine

Find the equation in slope-intercept form.

Answer to Problem 99RE

  $y=\left(\frac{-12}{7}\right)x-\frac{-24}{7}$ .

Explanation of Solution

Given information:

The given points are, $P\left(2,0\right),Q\left(-5,12\right)$ .

Calculation:

Consider the given points,

  $P\left(2,0\right),Q\left(-5,12\right)$

Line plotted between the points,

  

Now find slope of line to calculate the equation in slope-intercept form.

  $\begin{array}{l}y=mx+b\\ m=\frac{y_2-y_1}{x_2-x_1}=\frac{12-0}{-5-2}\\ m=\frac{-12}{7}\end{array}$

To find $y-\mathrm{int}ercept$ line passes through points $\left(2,0\right)$ .

  $\begin{array}{l}y=\frac{-12}{7}x+b\\ 0=\frac{-12}{7}\times 2+b\\ b=\frac{-24}{7}\left[\therefore y=mx+b\right]\\ y=\left(\frac{-12}{7}\right)x-\frac{-24}{7}\end{array}$

Hence, the equation in slope-intercept form is, $y=\left(\frac{-12}{7}\right)x-\frac{-24}{7}$

e.

To determine

Find the equation of this circle and draw the circle.

Answer to Problem 99RE

  ${\left(x-2\right)}^2+{\left(y\right)}^2=193$

  

Explanation of Solution

Given information:

The given points are, $P\left(2,0\right),Q\left(-5,12\right)$ .

Calculation:

Consider the given points,

  $P\left(2,0\right),Q\left(-5,12\right)$

The standard formula for circle has radius $r$ centred at $\left(a,b\right)$ centred point $\left(2,0\right)$ circle passes through the points $\left(-5,12\right)$ is,

Radius equal to distance between two points, $r=\sqrt{193}$

  $\begin{array}{l}{\left(x-a\right)}^2+{\left(y-b\right)}^2=r^2\\ r=\sqrt{193}\\ a=2,b=0\\ now,putthevalues,\\ {\left(x-2\right)}^2+{\left(y\right)}^2=193\end{array}$

  

Hence, the equation of circle is ${\left(x-2\right)}^2+{\left(y\right)}^2=193$ .