Single Choice

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$$\dfrac{\sqrt{5}-1}{\sqrt{5}+1}$$
B$$\dfrac{1-\sqrt{5}}{1+\sqrt{5}}$$
C$$\dfrac{1-\sqrt{7}}{1+\sqrt{7}}$$
Correct Answer
D$$\dfrac{\sqrt{7}-1}{\sqrt{7}+1}$$

Solution

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

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