Find the foot of the perpendicular through the point $A(-3;2)$ to line $2x-y+4=0$

Alissa Hutchinson
2022-05-07
Answered

Find the foot of the perpendicular through the point $A(-3;2)$ to line $2x-y+4=0$

You can still ask an expert for help

Athena Blanchard

Answered 2022-05-08
Author has **8** answers

Step 1

Any point on the line $2x-y+4=0$ is of the form $(t\in \mathbb{R})$

Then slope of the line $AN:{m}_{1}=\frac{(2t+4)-2}{t-(-3)}=\frac{2t+2}{t+3}$

Now slope of the line $2x-y+4=0:{m}_{2}=2$

Two lines are perpendicular, hence ${m}_{1}\cdot {m}_{2}=-1$

$\frac{2t+2}{t+3}\cdot 2=-1$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}t=\frac{-7}{5}$

Hence, $N=(\frac{-7}{5},\frac{6}{5})$

Any point on the line $2x-y+4=0$ is of the form $(t\in \mathbb{R})$

Then slope of the line $AN:{m}_{1}=\frac{(2t+4)-2}{t-(-3)}=\frac{2t+2}{t+3}$

Now slope of the line $2x-y+4=0:{m}_{2}=2$

Two lines are perpendicular, hence ${m}_{1}\cdot {m}_{2}=-1$

$\frac{2t+2}{t+3}\cdot 2=-1$

$\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}t=\frac{-7}{5}$

Hence, $N=(\frac{-7}{5},\frac{6}{5})$

Thaddeus Sanders

Answered 2022-05-09
Author has **1** answers

Step 1

We have $y=2x+4$ . Consider the line throudh A that is perpendicular to this line.

This line hence must have slope $-\frac{1}{2}$ .Thus its represented by $y=-\frac{1}{2}x+b$ for some b.

But since it passesthrough A, $2=\frac{3}{2}+b\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}b=\frac{1}{2}$ .

Thus the line is represented by $y=-\frac{1}{2}x+\frac{1}{2}$ . The foot of the perpendicular is simply the intersection of the two lines. Setting the y-coordinates, equal:

$2x+4=\frac{1}{2}-\frac{1}{2}x\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}4x+8=1-x\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}x=-\frac{7}{5}$

And then $y=-\frac{14}{5}+4=\frac{6}{5}$ . So the answer is $(-\frac{7}{5},\frac{6}{5})$ .

We have $y=2x+4$ . Consider the line throudh A that is perpendicular to this line.

This line hence must have slope $-\frac{1}{2}$ .Thus its represented by $y=-\frac{1}{2}x+b$ for some b.

But since it passesthrough A, $2=\frac{3}{2}+b\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}b=\frac{1}{2}$ .

Thus the line is represented by $y=-\frac{1}{2}x+\frac{1}{2}$ . The foot of the perpendicular is simply the intersection of the two lines. Setting the y-coordinates, equal:

$2x+4=\frac{1}{2}-\frac{1}{2}x\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}4x+8=1-x\phantom{\rule{thickmathspace}{0ex}}\u27f9\phantom{\rule{thickmathspace}{0ex}}x=-\frac{7}{5}$

And then $y=-\frac{14}{5}+4=\frac{6}{5}$ . So the answer is $(-\frac{7}{5},\frac{6}{5})$ .

asked 2020-12-28

What is geometry?

asked 2021-12-21

A. Let $(X,\text{}d)$ be a metric space. Define a diameter of a subset A of X and then the diameter of an open ball with center at $x}_{0$ and radios $r>0$

B. Use (A) to show that every convergence sequence in$(X,\text{}d)$ is bounded.

B. Use (A) to show that every convergence sequence in

asked 2021-02-16

Consider the line represented by the equation

asked 2021-11-29

What is the volume of this cone?

Use$\pi \approx 3.14$ and round your answer to the nearest hundredth.

What is the volume of this cylinder?

Use$\pi \approx 3.14$ and round your answer to the nearest hundredth.

Use

What is the volume of this cylinder?

Use

asked 2021-11-28

What is the Volume of this sphere?

Use$\pi \approx 3.14$ and round your answer to the nearest hundredth,

Maple syrup in a bottle has a mass of 918 grams and a density of 1.36 grams per cubic centimeter. What is its volume?

Use

Maple syrup in a bottle has a mass of 918 grams and a density of 1.36 grams per cubic centimeter. What is its volume?

asked 2021-12-14

Find $m\mathrm{\angle}F$

asked 2021-12-09

How to construct corresponding angles?