Question

If two lines drawn from a point $P(2,3)$ intersect the line $x+y=6$ at a distance $\sqrt{\dfrac{2}{3}}$, then the angle between the lines is

Hint:

For such geometry problems, remember the standard angle formula using perpendicular distance.

Answer 1

Correct Answer: 1

Step 1: Find perpendicular distance of point P from the line

Given point P(2, 3) and line x + y = 6

Distance,

d = |2 + 3 − 6| / √(1² + 1²)

d = 1 / √2

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Step 2: Interpret the geometric condition

The two lines from point P intersect the line x + y = 6 at points which are at a distance

r = √(2/3)

from P.

Thus, these intersection points lie on a circle with center P and radius r.

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Step 3: Use angle subtended by a chord

Let θ be the angle between one of the lines and the perpendicular from P to the given line.

Then,

cos θ = d / r

cos θ = (1/√2) / √(2/3)

cos θ = (1/√2) × (√3/√2)

cos θ = √3 / 2

θ = 30°

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Step 4: Find the angle between the two lines

Angle between the two lines = 2θ

= 2 × 30°

= 60°

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Final Answer:

The angle between the two lines is 60°