Question

Find the equations to the straight lines which pass through the point (h, k) and are inclined at angle $ta{n}^{-1}m$ to the straight line $y=mx+c$.

Answer 1

We know that the equations of two lines passing through a point $(x_1,y_1)$ and making an angle $\alpha$ with the given line $y=m^{{}^{\prime }}x+c$ are

$y-y_1=\frac{m^{{}^{\prime }}\pm tan\alpha }{1\pm m^{{}^{\prime }}tan\alpha }(x-x_1)$

Here,

$x_1=h$, $y_1=k$, $\alpha =ta{n}^{-1}m$, $m^{{}^{\prime }}=m$

So, the equations of the required lines are

$y-k=\frac{m+m}{1-m^2}(x-h)$ and

$y-k=\frac{m-m}{1+m^2}(x-h)$

$\Rightarrow y-k=\frac{2m}{1-m^2}(x-h)$ and $y-k=0$

$\Rightarrow (y-k)(1-m^2)=2m(x-h)$ and $y=k$