Find the direction in which a straight line must be drawn through the point (1,2), so that it's point of - Maths - Straight Lines

Question

Find the direction in which a straight line must be drawn
through the point (1,2), so that it's point of intersection with
the line x+y=4 is at distance sqrt(2/3) from
the given point (pk LII q-8)

Aakash Sharma · Accepted Answer

Let P = (1, 2)

and Let AB be the given line

x + y = 4 ......... (1)

Let the line through P making an angle Q with x axis cuts the line AB at Q at a distance $\sqrt{\frac{2}{3}}$ from P.

Then

$Q = (1 + \sqrt{\frac{2}{3}} \cos θ, 2 - \sqrt{\frac{2}{3}} \sin θ)$

Since Q lies on Line (1)

$∴ 1 + \sqrt{\frac{2}{3}} \cos θ + 2 + \sqrt{\frac{2}{3}} \sin θ = 4 \Rightarrow \cos θ + \sin θ = \sqrt{\frac{3}{2}} \Rightarrow \frac{1}{\sqrt{2}} \cos θ + \frac{1}{\sqrt{2}} \sin θ = \frac{\sqrt{3}}{2} \Rightarrow \cos 45 ° \cos θ + \sin 45 ° \sin θ = \frac{\sqrt{3}}{2} \Rightarrow \cos (θ - 45 °) = \cos 30 ° \Rightarrow θ - 45 ° = 2 n π \pm 30 °, n = θ \pm 1, \pm 2$

⇒ θ = 15°, 75° (taking only values between 0° and 180°)

-5

Find the direction in which a straight line must be drawn through the point (1,2), so that it's point of intersection with the line x+y=4 is at distance sqrt(2/3) from the given point.(pk LII q-8)

Find the direction in which a straight line must be drawn through the point (1,2), so that it's point of intersection with the line x+y=4 is at distance sqrt(²/₃) from the given point.(pk LII q-8)