Question:

If p be the length of the perpendicular from the origin on the straight line x + 2by = 2p ,then what is the value of b?

Updated On: Jul 6, 2022

$\frac{1}{p}$
p
$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$

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The Correct Option is D

Solution and Explanation

Length of perpendicular from the origin on the straight line x + 2by - 2p = 0 is $\left|\frac{0+2b\times0-2p}{\sqrt{1^{2}+\left(2b\right)^{2}}} \right| = p $ or $p= \left|\frac{-2p}{\sqrt{1^{2}+4b^{2}}}\right| $ or $p^{2} = \frac{4p^{2}}{1+4b^{2}} $ $ \Rightarrow \frac{4}{1+4b^{2}} = 1 $ $ \Rightarrow 1+4b^{2}=4 $ or $4b^{2} = 3 \Rightarrow b^{2} = \frac{3}{4}$ $ \Rightarrow b = \pm \frac{\sqrt{3}}{2} \Rightarrow b = \frac{\sqrt{3}}{2} $ matches with the given option.

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Notes on Straight lines

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