Question

The equation of the straight line joining the origin to the point of intersection of $y-x+7=0\text{ and }y+2x-2=0$ is
(a) $3x+4y=0$
(b) $3x-4y=0$
(c) $4x-3y=0$
(d) \[4x+3y=0\]

Answer 1

Hint: Assume that the equation of the line in the slope-intercept form, passing through the origin is $y=mx$, where $m$ is the slope of the line, that is, inclination of the line with respect to the x-axis. Solve the given two equations and determine the coordinates of the point of intersection. Now, substitute the value of obtained coordinates in the assumed equation of the line to get the slope.

Complete Step-by-Step solution:
Let us assume the equation of the line in the is $y=mx$. Here, the intercept $c$ is zero because it is given that the line is passing through origin.
Now, we have been provided with two equations: $y-x+7=0\text{ and }y+2x-2=0$.
$\begin{align}
  & y-x+7=0.........................(i) \\
 & y+2x-2=0.......................(ii) \\
\end{align}$
Let us solve these equations to get the coordinates of the point of intersection. Therefore, subtracting equation (ii) from equation (i), we get,
$\begin{align}
  & -3x+9=0 \\
 & 3x=9 \\
 & \Rightarrow x=3 \\
\end{align}$
Substituting $x=3$ in equation (i), we get,
$\begin{align}
  & y-3+7=0 \\
 & y+4=0 \\
 & \Rightarrow y=-4 \\
\end{align}$
Therefore, the coordinates of the point of intersection is $\left( 3,-4 \right)$.
Now, we know that slope of a line is $m=\dfrac{\Delta y}{\Delta x}=\dfrac{{{y}_{1}}-{{y}_{2}}}{{{x}_{1}}-{{x}_{2}}}$.
Substituting, ${{y}_{1}}=0,{{y}_{2}}=-4,{{x}_{1}}=0\text{ and }{{x}_{2}}=3$, we get,
$m=\dfrac{0-(-4)}{0-3}=\dfrac{4}{-3}=\dfrac{-4}{3}$
Now, substituting the value of $m$ in the equation of the line, we get,
$\begin{align}
  & y=\dfrac{-4}{3}x \\
 & \Rightarrow 3y=-4x \\
 & \Rightarrow 3y+4x=0 \\
\end{align}$
Hence, option (d) is the correct answer.

Note: One may note that we have assumed the required equation of line as $y=mx$. Here, $c$ is not present because the line passes through the origin. You may assume the equation as $y=mx+c$ and when you will solve the equation for the value of $c$ then you will get $c=0$. So, in this way you can verify that whenever a line passes through origin then the simplified form of the equation of line is $y=mx$.