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

Updated On: June 02, 2025
  • $\frac{\sqrt{5}-1}{\sqrt{5}+1}$
  • $\frac{1- \sqrt{5}}{1+ \sqrt{5}}$
  • $\frac{1- \sqrt{7}}{1+ \sqrt{7}}$
  • $\frac{\sqrt{7}-1}{\sqrt{7}+1}$
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The Correct Option is C

Solution and Explanation

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

Straight lines

A straight line is a line having the shortest distance between two points. 

A straight line can be represented as an equation in various forms,  as show in the image below:

 

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y0 = m (x – x0)

2. Two – Point Form

Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2)  are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes

The slope of P2P = The slope of P1P2 , i.e.

\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)

Hence, the equation becomes:

y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c