Let the slope of the required line be m1.
The given line can be written as y=12x−32, which is of the form y=mx+c
∴ slope of the given line =m2=12
It is given that the angle between the required line and line x−2y=3 is 45∘
We
know that if θ is the acute angle between lines
l1 and l2 with slopes m1 and
m2 respectively, then tanθ=∣∣∣m2−m11+m1m2∣∣∣
∴tan45∘=∣∣∣m2−m11+m1m2∣∣∣
⇒1=∣∣
∣∣12−m11+m12∣∣
∣∣
⇒1=∣∣
∣∣(1−2m12)2+m12∣∣
∣∣
⇒1=±∣∣∣1−2m12+m1∣∣∣
⇒m1=−13orm1=3
case 1; m1=3
The equation of the line passing through (3,2) and having a slope of 3 is
y−2=3(x−3)⇒3x−y=7
case 2: m1=−13
The equation of the line passing through (3,2) and having a slope of −13 is.
y−2=−13(x−3)
⇒3y−6=x+3⇒x+3y=9