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.
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 $\displaystyle y=mx+b$ where b is the y-intercept
$\displaystyle 2=-4m+b$, now the y-intercept can be 2 or -2 for the two lines...
$\displaystyle 2=-4m+2$
$\displaystyle m=0$
$\displaystyle b=2$
$\displaystyle 2=-4m-2$
$\displaystyle m=-1$
$\displaystyle b=-2$
You have everything you need!
The equation of the line with the slope $\displaystyle m$ and passing through the point $\displaystyle M_0(x_0,y_0) is$
$\displaystyle y=y_0=m(x-x_0)$
In this case $\displaystyle y-2=m(x+4)$ or $\displaystyle mx-y+4m+2=0$
The distance from the point $\displaystyle M_0(x_0,y_0)$ to the line $\displaystyle ax+by+c=0$ is
$\displaystyle \displaystyle d=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$
In this case $\displaystyle \displaystyle d=\frac{|4m+2|}{\sqrt{m^2+1}}=2\Rightarrow |2m+1|=\sqrt{m^2+1}$
Squaring both members and solving the quadratic we get
$\displaystyle \displaystyle m_1=0, \ m_2=-\frac{4}{3}$