# Thread: Studying for exam need help!

1. ## Studying for exam need help!

Need help with this question from coordinate geometry:

Write down the equation of any line through the point (-4, 2).
Hence find the equations of the two lines through the point (-4, 2) whose perpendicular distance from the origin is 2.

2. Originally Posted by milan
Need help with this question from coordinate geometry:

Write down the equation of any line through the point (-4, 2).
Hence find the equations of the two lines through the point (-4, 2) whose perpendicular distance from the origin is 2.
Use $y=mx+b$ where b is the y-intercept

$2=-4m+b$, now the y-intercept can be 2 or -2 for the two lines...

$2=-4m+2$
$m=0$
$b=2$

$2=-4m-2$
$m=-1$
$b=-2$

You have everything you need!

3. The equation of the line with the slope $m$ and passing through the point $M_0(x_0,y_0) is$
$y=y_0=m(x-x_0)$
In this case $y-2=m(x+4)$ or $mx-y+4m+2=0$

The distance from the point $M_0(x_0,y_0)$ to the line $ax+by+c=0$ is
$\displaystyle d=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$
In this case $\displaystyle d=\frac{|4m+2|}{\sqrt{m^2+1}}=2\Rightarrow |2m+1|=\sqrt{m^2+1}$
Squaring both members and solving the quadratic we get
$\displaystyle m_1=0, \ m_2=-\frac{4}{3}$

4. One of our solutions is correct depending on the wording of the question.