<s>[INST] Two lines passing through $(-2,5)$ making angles $\arctan\frac 34$ & $\arctan\frac 23$ with the line $x-y+5=0$.Find the equations of the lines Two straight lines pass through the point $(-2,5)$ such that one of them makes an angle of $\arctan \frac 34$
with the given line $x-y+5=0$
and the given line makes an angle of $\arctan \frac 23$
with the other line . Find the equations of the two lines.
$\bf{Try} :$
Let the equation of the line be $y=mx+c$.$\cdots (1)$
If $(1)$ makes angle $\theta$ with the given line then $\tan\theta =| \frac {m-1}{1+m}|$
So by the given condition we have
$| \frac {m-1}{1+m}|=\frac 34$
Solving we get $m=7,\frac 17$
Putting these values of $m$ and using the condition that the line passes through $(-2,5)$ we get two lines as
$y=7x+19  \cdots (2)$
$7y=x+37\cdots (3)$
Similarly $| \frac {m-1}{1+m}|=\frac 23$
gives $m=5,\frac 15$ and we get the equations of the lines as
$y=5x+15\cdots (4)$
and $5y=x+27\cdots (5)$
But the answer is given as the lines $(2)$ and $(5)$. Why the lines $(3)$ and $(4)$ are excluded?
 [/INST] {Red (2) Orange (3) Violet (4) Grey(5)} all for image if I will be able to upload image.
If question strictly wants positive angle for  statement then:
(3) makes $-\tan^{-1}(\frac {3}{4})$ with $x-y+5=0$
Similarly (4) makes $-\tan^{-1}(\frac{2}{3})$ with $x-y+5=0$
You may draw graph to check.
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