Question

The line through point $(m,-9)$ and $(7,m)$ has slope m. The y-intercept of this line, is?

• A. $-18$
• B. $-6$
• C. $6$
• D. $18$

Answer 1

The correct option is A $-18$

We know that the slope $m$ of a line passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$.

Let $(m,-9)=(x_1,y_1)$ and $(7,m)=(x_2,y_2)$, then the slope of the line can be determined as:

$m=\frac{y_2-y_1}{x_2-x_1}\Rightarrow m=\frac{m-(-9)}{7-m}\Rightarrow m=\frac{m+9}{7-m}\Rightarrow m(7-m)=m+9$

$\Rightarrow {-m}^2+7m=m+9\Rightarrow m^2-7m+m+9=0\Rightarrow m^2-6m+9=0\Rightarrow {(m-3)}^2=0\Rightarrow m-3=0\Rightarrow m=3$

Therefore, the points are $(3,-9)$ and $(7,3)$.

We also know that the equation of the line passing through a point $(x_1,y_1)$ is $(y-y_1)=m(x-x_1)$ where $m$ is the slope of the line.

Thus, the equation of the line passing through the point $(7,3)$ is as follows:

$(y-y_1)=m(x-x_1)\Rightarrow (y-3)=3(x-7)\Rightarrow y-3=3x-21\Rightarrow 3x-21-y+3=0\Rightarrow 3x-y-18=0\Rightarrow 3x-y=18$

Now, put $x=0$ in the above equation to get the y-intercept as:

$(3\times 0)-y=18\Rightarrow 0-y=18\Rightarrow y=-18$

Hence, the y-intercept of the line is $y=-18$.