Question

Which point on the y-axis lies on the line that passes through point C and is perpendicular to line AB?

Answer 1

The line perpendicular to the line AB passes through the point $(0,2)$ i.e., $\fbox{\begin\\\\ \bf option 3\\\end{minispace}}$.

Further explanation:

From the given figure in the question it is observed that the line AB passes through the points $(-2,4)$ and $(2,-8)$.

The coordinate for the point C is $(6,4)$.

Step1: Obtain the slope of the line AB.

The slope of a line which passes through the points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is calculated as follows:

$\fbox{\begin\\\ \math m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\\\end{minispace}}$  (1)

It is given that the line AB passes through the points $(-2,4)$ and $(2,-8)$.

To obtain the slope for the line AB substitute $-2$ for $x_{1}$, $2$ for $x_{2}$, $4$ for $y_{1}$ and $-8$ for $y_{2}$ in equation (1).

$\begin{aligned}m&=\frac{-8-4}{2+2}\\&=\frac{-12}{4}\\&=-3\end{aligned}$

Therefore, the slope of the line AB is $-3$.  

Consider the slope of AB as, $m_{1}$ so, $m_{1}=-3$.

Step2: Obtain the slope of the perpendicular line.  

The slope of line AB is $m_{1}=-3$.

Consider a line which is perpendicular to the line AB passing through the point C. Assume the slope of the perpendicular line as $m_{2}$.

The product of slope of two mutually perpendicular lines is always equal to $-1$.

The equation formed for the slope is as follows:  

$\fbox{\begin\\\ \math m_{1}\times m_{2}=-1\\\end{minispace}}$

Substitute the value of $m_{1}$ in the above equation.

$\begin{aligned}-3\times m_{2}&=-1\\m_{2}&=\frac{1}{3}\end{aligned}$

Therefore, the slope of the perpendicular line is $m_{2}=\frac{1}{3}$.

Step3: Obtain the equation of the perpendicular line.

The slope for perpendicular line is $\frac{1}{3}$ and the line passes through the point C. The coordinate for the point C are $(6,4)$.

The point slope form of a line is as follows:

$\fbox{\begin\\\ \math (y-y_{1})=m(x-x_{1})\\\end{minispace}}$

Substitute $\frac{1}{3}$ for $m$, $6$ for $x_{1}$ and $4$ for $y_{1}$ in the above equation.

$\begin{aligned}(y-4)&=\frac{1}{3}(x-6)\\3y-12&=x-6\\3y-x&=6\end{aligned}$

Therefore, the equation of the perpendicular line is $3y-x=6$.

Option 1:

In option 1 it is given that the line perpendicular to AB passes through the point $(-6,0)$.  

The equation of the line which is perpendicular to AB is $3y-x=6$.

Substitute $-6$ for $x$ in the above equation.

$\begin{aligned}3y-(-6)&=6\\3y+12&=6\\3y&=-6\\y&=-2\end{aligned}$

From the above calculation it is concluded that the line passes through the point $(-6,-2)$.

This implies that option 1 is incorrect.

Option 2:

In option 2 it is given that the line perpendicular to AB passes through the point $(0,-6)$.

The equation of the line which is perpendicular to AB is $3y-x=6$.

Substitute $0$ for $x$ in the above equation.

$\begin{aligned}3y-(0)&=6\\3y&=6\\y&=2\end{aligned}$

From the above calculation it is concluded that the line passes through the point $(0,2)$.

This implies that option 2 is incorrect.

Option 3:

In option 3 it is given that the line perpendicular to AB passes through the point $(0,2)$.

The equation of the line which is perpendicular to AB is $3y-x=6$.

Substitute $0$ for $x$ in the above equation.

$\begin{aligned}3y-(0)&=6\\3y&=6\\y&=2\end{aligned}$

From the above calculation it is concluded that the line passes through the point $(0,2)$.

This implies that option 3 is correct.

Option 4:

In option 4 it is given that the line perpendicular to AB passes through the point $(2,0)$.

The equation of the line which is perpendicular to AB is $3y-x=6$.

Substitute $2$ for $x$ in the above equation.

$\begin{aligned}3y-(2)&=6\\3y&=8\\y&=\frac{8}{3}\end{aligned}$

From the above calculation it is concluded that the line passes through the point $(2,\frac{8}{3})$.

This implies that option 4 is incorrect.

Therefore, the line perpendicular to the line AB passes through the point $(0,2)$ i.e., $\fbox{\begin\\\ \bf option 3\\\end{minispace}}$.

Learn more:

1. A problem on composite function brainly.com/question/2723982  

2. A problem to find radius and center of circle brainly.com/question/9510228  

3. A problem to determine intercepts of a line brainly.com/question/1332667  

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Lines

Keywords: Geometry, coordinate geometry, lines, equation, graph, curve, slope, perpendicular, point slope form, slope intercept form.

Answer 2

We know that

if two lines are perpendicular m1*m2=-1


Line AB
A(-2,4)
B(2,-8)
m1=(y2-y1)/(x2-x1)---------> m1=(-8-4)/(2+2)-----> m1=-12/4-----> -3

m2=1/3

then
with m2=1/3 and the point C( 6,4)
y-4=(1/3)*(x-6)--------> y-4=(x/3)-2------> y=(x/3)+2

the y intercept is for x=0
then
 y=(x/3)+2------>  y=(0/3)+2------> y=2
the y intercept is the point (0,2)

using a graph tool
see the attached figure



the answer is 
the point (0,2)