Question

The straight line passing through the point of intersection of the straight lines $x - 3y + 1 = 0$ and $2x + 5y - 9 = 0$ and having infinite slope and at a distance of $2$ units from the origin, has the equation
(1) $x = 2$
(2) $3x + y - 1 = 0$
(3) $y = 1$
(4) None of these

Answer 1

Hint : A straight line is an endless one dimensional figure that has no width. The straight line is a combination of endless points joined on both sides of a point. A straight line does not have any curve in it. The straight line can be horizontal, vertical or slanted. When two or more straight lines cross each other in a plane, they are called intersecting lines.

Complete step-by-step answer :
First we find the intersection point of the straight lines
Take $x - 3y + 1 = 0$ ……………………………..(1)
$2x + 5y - 9 = 0$ …………………………………..(2)
From equation (1) , we get
$x = 3y - 1$ ………………………………(3)
Use equation (3) in equation (2) , we get
$2(3y - 1) + 5y - 9 = 0$
Multiplying and we get
$\Rightarrow 6y - 2 + 5y - 9 = 0$
$\Rightarrow 11y - 11 = 0$
$\Rightarrow 11y = 11$
We divide both sides by $11$ , we get
$\Rightarrow \dfrac{{11y}}{{11}} = \dfrac{{11}}{{11}}$
$\Rightarrow y = 1$
Put the value of $y = 1$ in equation (3) , we get
$x = 3 \times 1 - 1$
$\Rightarrow x = 3 - 1$
$\Rightarrow x = 2$
Therefore the intersection point is $(2,1)$ .
From the given data we have infinite slope , means $\theta = \dfrac{\pi }{2}$
Therefore the straight line parallel to Y-axis i.e., $x = a$ , where $a$ is the distance from origin.
And also given that the distance from origin is $2$ units.
Then we have the straight line $x = 2$ .
Therefore the option (1) is correct.
So, the correct answer is “Option 1”.

Note : The slope of a line is a number measured as its “steepness”, usually denoted by the letter $m$ . It is the change in $y$ for a unit change in $x$ for a change in $x$ along the line. The formula of slope of a straight line is $m = \tan \theta$ , where $\theta$ is the angle between the X-axis and the straight line.