Question:

If $P_1$ and $P_2$ be the length of perpendiculars from the origin upon the straight lines $x \sec \theta + y cosec \theta = a$ and $x \cos \theta - y \sin \theta = a \cos 2 \theta$ respectively, then the value of $4P_1{^2} + P_2{^2}$.

Updated On: May 19, 2022
  • $a^2$
  • $ 2 a^2$
  • $a^2/ 2$
  • $3a^2$
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The Correct Option is A

Solution and Explanation

We have $P_1$ = length of perpendicular from $(0, 0)$ on
$x \sec \theta + y \cos ec \theta = a$
i.e. $P_{1} = \left|\frac{a}{\sqrt{\sec^{2} \theta +\cos ec^{2} \theta}}\right| = \left|a \sin\theta \cos\theta\right| $
$ = \left|\frac{a}{2} \sin 2\theta\right| 2P_{1} = \left|a \sin 2 \theta\right|$
$P_2$= Length of the perpendicular from $(0, 0) $ on
$ x \cos \theta -y \sin\theta =a \cos2\theta $
$ P_{2} = \left|\frac{a \cos 2 \theta}{\sqrt{\cos^{2} \theta+\sin^{2} \theta}}\right| = \left|a \cos2\theta\right| $
Now, $ 4P_{1} ^{2} + P_{2 } ^{2}= a^{2} \sin^{2} 2 \theta + a^{2} \cos^{2} 2\theta = a^{2}. $
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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