Question

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

• A. $\frac{\sqrt{5}-1}{\sqrt{5}+1}$
• B. $\frac{1-\sqrt{5}}{1+\sqrt{5}}$
• C. $\frac{1-\sqrt{7}}{1+\sqrt{7}}$
• D. $\frac{\sqrt{7}-1}{\sqrt{7}+1}$

Answer 1

Answer: (C)

$x=2+rcos\theta$
$y=3+rsin\theta$
$\Rightarrow$ $2+rcos\theta +3+rsin\theta =7$
$\Rightarrow$ $r(cos\theta +sin\theta$) = 2
$\Rightarrow sin\theta +cos\theta =\frac{2}{r}=\frac{2}{\pm 4}=\pm \frac{1}{2}$
$\Rightarrow 1+sin2\theta =\frac{1}{4}$
$\Rightarrow sin2\theta =-\frac{3}{4}$
$\Rightarrow \frac{2m}{1+m^2}=-\frac{3}{4}$
$\Rightarrow 3m^2+8m+3=0$
$\Rightarrow m=\frac{-4\pm \sqrt{7}}{1-7}$
$\frac{1-\sqrt{7}}{1+\sqrt{7}}=\frac{(1-\sqrt{7}{)}^2}{1-7}=\frac{8-2\sqrt{7}}{-6}=\frac{-4+\sqrt{7}}{3}$