Question

The equation of a line through the intersection of lines \[x=0\] and \[y=0\] and through the point \[\left( 2,2 \right)\] is
A. $y=x-1$
B. $y=-x$
C. $y=x$
D. $y=-x+2$

Answer 1

Hint: In this question we are to find the equation of the line that passes through the point of intersection of the given lines and a point \[\left( 2,2 \right)\]. For this, the formula for the equation of a line that passes through two points. Then, by calculating, we get the required equation.



Formula Used:The equation of the line passing through $({{x}_{1}},{{y}_{1}})$ and $({{x}_{2}},{{y}_{2}})$ is
$y-{{y}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}(x-{{x}_{1}})$
Where $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ is said to be the slope of the line.



Complete step by step solution:It is given that; the required equation of the line is passing through the intersection of the given lines. That are \[x=0\] and \[y=0\].
I.e., x-axis and y-axis. So, they intersect at the origin.
So, the point of intersection between these lines is $(0,0)$
Thus, the required equation of the line passes through the points $(0,0)$ and \[\left( 2,2 \right)\].
So, consider $({{x}_{1}},{{y}_{1}})=(0,0);({{x}_{2}},{{y}_{2}})=(2,2)$
Then, by using the formula of the equation of the line with two points, we get
$\begin{align}
  & y-{{y}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}(x-{{x}_{1}}) \\
 & \text{ }y-0=\dfrac{2-0}{2-0}(x-0) \\
 & \text{ }y=1(x-0) \\
 & \text{ }\therefore y=x \\
\end{align}$
Therefore, the required equation of the line is $y=x$



Option ‘C’ is correct

Note: Here we may go wrong with the intersection of the given lines. Since the given lines in this question are axes itself, it became easy to calculate the point of intersection. Otherwise, we can use the substitution method to find the point of intersection. So, the equation of a line with two points is calculated by substituting in the equation of the line with two points formula. We can also use point-slope formula by calculating the slope with those two points.