Question

The equation of a line through the intersection of lines $x=0$ and $y=0$ and through the
point $(2,2)$ is
(a) $y=x-1$
(a) $y=-x$
(b) $y=x$
(c) $y=-x+2$

Answer 1

Hint: Substitute the given points into the standard equation of line formula.

The equations of the given lines are $x=0$ and $y=0$. The coordinates of the point of
intersections are already available. This means that the point of intersection of these lines would be
$(0,0)$. Another point through which the required line passes through is given in the
question as $(2,2)$.
Now, we know that the equation of a line passing through two points $$(x_1,y_1)$$ and
$$(x_2,y_2)$$ is given by,
$$\begin{array}{rl}& y-y_1=m(x-x_1)\\ & \Rightarrow y-y_1={\displaystyle \frac{y_2-y_1}{x_2-x_1}}\times (x-x_1)\end{array}$$
So, the equation of the required line passing through $(0,0)$ and $(2,2)$ can
be obtained as,
$y-0={\displaystyle \frac{2-0}{2-0}}(x-0)$
Therefore, the equation of the required line is $y=x$.
Hence, we get option (c) as the correct answer.

Note: The equations $x=0$ and $y=0$ indicate that the required line passes through the origin. So,
the formula to find the equation of the line passing through a point $(x_1,y_1)$
can be obtained as $y=mx$, where $m={\displaystyle \frac{y_1}{x_1}}$ is the slope.