We have given a point (2, 3), through which two lines are passing and intersects at an angle of 60°.
Let m be the slope of the other line
⇒ tan 60° = |

m−2

1+2m

| ⇒ ± √3 =

m−2

1+2m

⇒ √3 =

m−2

1+2m

... (i)
or - √3 =

m−2

1+2m

... (ii)
⇒ √3+2√3m = m - 2 or √3+2√3m = 2 - m
⇒ √3 + 2 = m - 2√3 m or 2√3 m + m = 2 - √3
⇒ m =

√3+2

1−2√3

or m =

2−√3

1+2√3

Since the line passes through (2, 3)
∴ Its equation is either
y - 3 =

√3+2

1−2√3

(x−2) ... (iii)
or y - 3 =

2−√3

1+2√3

(x−2) ... (iv)
(iii) ⇒ (y − 3)(1 − 2√3) = (√3 + 2) (x − 2)
⇒ y (1 − 2√3) − 3(1 − 2√3) = (√3 + 2)x − 2(√3 + 2)
⇒ y (1 − 2√3) − (3 − 6 √3) = (√3 + 2)x − 2 √3 −4
⇒ 2 √3 + 4 − 3 + 6 √3 =(√3 + 2)x +(2√3 − 1)y
⇒(√3 + 2)x + (2√3 − 1)y− 8 √3 − 1 = 0 ... (v)
And
(iv) ⇒ (y − 3) (1 + 2√3) = (2 − √3) (x −2)
⇒ y(1 + 2√3) − 3 (1 + 2√3) = (2 − √3)x − 2(2 − √3)
⇒ − 3 − 6 √3 + 2(2 − √3) = (2 − √3)x− y(1+ 2 √3)
⇒ − 3 − 6 √3 +4 − 2 √3 =(2 − √3)x − y (1 + 2 √3)
⇒ (2 − √3)x −(1 + 2 √3)y + 8√3 − 1 = 0 ... (vi)
Thus (v) and (vi) are the required equations of the line.