Slope of a line passing through P(2,3) and intersecting the line x+y=7 at a distance of 4 units from P, is?

The correct option is (1-√7)/(1+√7)
x = 2 + rcosθ
y = 3 + rsinθ
=> 2 + rcosθ + 3 + rsinθ = 7
=> r(cosθ + sinθ) = 2
Given r = 4
=> sinθ + cosθ = 2/4 = 1/2
Squaring both sides,
=> (sinθ + cosθ)² = (1/2)²
=> sin²θ + cos²θ + 2sinθcosθ = 1/4
=> 1 + 2sinθcosθ = 1/4
=> 2sinθcosθ = -3/4
=> sin2θ = -3/4
Let m1 be the slope of the line x + y = 7, and m2 be the slope of the line passing through P(2,3).
m1 = -1
The angle between the lines is given by:
tanθ = |(m2 - m1)/(1 + m1m2)|
Since sin2θ = -3/4, we have:
2m1m2/(1 + m1²)(1 + m2²) = -3/4
2(-m2)/(1 + m2²) = -3/4
8(-m2) = -3(1 + m2²)
-8m2 = -3 -3m2²
3m2² - 8m2 + 3 = 0
Solving the quadratic equation for m2:
m2 = (8 ± √(64 - 36))/6 = (8 ± √28)/6 = (8 ± 2√7)/6 = (4 ± √7)/3
m2 = (4 + √7)/3 or m2 = (4 - √7)/3
Let's consider the line passing through (2,3) with slope m2.
y - 3 = m2(x - 2)
If m2 = (4 + √7)/3, then y - 3 = (4 + √7)/3 (x - 2)
If m2 = (4 - √7)/3, then y - 3 = (4 - √7)/3 (x - 2)
The equation of the line passing through (2,3) and having a slope of (1-√7)/(1+√7) is given by:
y - 3 = [(1 - √7)/(1 + √7)](x - 2)
Therefore, the slope is (1 - √7)/(1 + √7).