Question

Find the equation of the line through the point(3, 2) which makes an angle of 45° with the line x - 2y = 3 ?

(Kerala PET 2007)​

Answer 1

Step-by-step explanation:

Let the slope of the required line be m

1

.

The given line can be written as y=

2

1

x−

2

3

, which is of the form y=mx+c

∴ slope of the given line =m

2

=

2

1

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 l

1

and l

2

with slopes m

1

and m

2

respectively, then tanθ=

∣

∣

∣

∣

∣

1+m

1

m

2

m

2

−m

1

∣

∣

∣

∣

∣

∴tan45

∘

=

∣

∣

∣

∣

∣

1+m

1

m

2

m

2

−m

1

∣

∣

∣

∣

∣

⇒1=

∣

∣

∣

∣

∣

1+

2

m

1

2

1

−m

1

∣

∣

∣

∣

∣

⇒1=

∣

∣

∣

∣

∣

∣

2

2+m

1

(

2

1−2m

1

)

∣

∣

∣

∣

∣

∣

⇒1=±

∣

∣

∣

∣

∣

2+m

1

1−2m

1

∣

∣

∣

∣

∣

⇒m

1

=−

3

1

orm

1

=3

case 1; m

1

=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: m

1

=−

3

1

The equation of the line passing through (3,2) and having a slope of −

3

1

is.

y−2=−

3

1

(x−3)

⇒3y−6=x+3⇒x+3y=9