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.