# 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) $y=x-1$

(a) $y=-x$

(b) $y=x$

(c) $y=-x+2$

Last updated date: 28th Mar 2023

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Total views: 312k

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Answer

Verified

312k+ views

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

$\left( 0,0 \right)$. Another point through which the required line passes through is given in the

question as $\left( 2,2 \right)$.

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{align}

& y-{{y}_{1}}=m(x-{{x}_{1}}) \\

& \Rightarrow y-{{y}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\times (x-{{x}_{1}}) \\

\end{align}\]

So, the equation of the required line passing through $\left( 0,0 \right)$ and $\left( 2,2 \right)$ can

be obtained as,

$y-0=\dfrac{2-0}{2-0}\left( x-0 \right)$

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 $\left( {{x}_{1}},{{y}_{1}} \right)$

can be obtained as $y=mx$, where $m=\dfrac{{{y}_{1}}}{{{x}_{1}}}$ is the slope.

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

$\left( 0,0 \right)$. Another point through which the required line passes through is given in the

question as $\left( 2,2 \right)$.

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{align}

& y-{{y}_{1}}=m(x-{{x}_{1}}) \\

& \Rightarrow y-{{y}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\times (x-{{x}_{1}}) \\

\end{align}\]

So, the equation of the required line passing through $\left( 0,0 \right)$ and $\left( 2,2 \right)$ can

be obtained as,

$y-0=\dfrac{2-0}{2-0}\left( x-0 \right)$

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 $\left( {{x}_{1}},{{y}_{1}} \right)$

can be obtained as $y=mx$, where $m=\dfrac{{{y}_{1}}}{{{x}_{1}}}$ is the slope.

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