Q.A line passes through $(2,2)$ and is perpendicular to the line $3x+y=3$. Its $y$-intercept is
(A) $\dfrac{1}{3}$
(B) $\dfrac{2}{3}$
(C) $1$
(D) $\dfrac{4}{3}$

The key idea here is the Perpendicular Slopes Condition.

First, we find the slope of the given line $3x+y=3$. Rewriting it in slope-intercept form $y=mx+c$, we get $y=-3x+3$. The slope of this line, $m_1$, is $-3$.

For two lines to be perpendicular, the product of their slopes must be $-1$. If the slope of the required line is $m_2$, then $m_1 \cdot m_2 = -1$.

So, $(-3) \cdot m_2 = -1$, which gives $m_2 = \frac{1}{3}$.

Now we have the slope $m_2 = \frac{1}{3}$ and a point $(2,2)$ through which the line passes. Using the point-slope form $y-y_1 = m(x-x_1)$:

$y-2 = \frac{1}{3}(x-2)$

$3(y-2) = x-2$ …

When two lines are perpendicular, their slopes have a special relationship. This relationship is fundamental to solving problems involving perpendicular lines. Geometrically, perpendicular lines intersect at a right angle ($90^\circ$). Algebraically, if one line has a slope $m_1$, then any line perpendicular to it will have a slope $m_2$ such that their product is $-1$. That is, $m_1 \cdot m_2 = -1$, or $m_2 = -\frac{1}{m_1}$. This is often called the negative reciprocal condition.

We are given a line $3x+y=3$ and asked to find the y-intercept of a new line that passes through $(2,2)$ and is perpendicular to the given line. Our strategy will be to first find the slope of the given line, then use the perpendicularity condition to find the slope of the new line. With the slope and a point on the new line, we can write its equation and then easily find its y-intercept.

1. Find the slope of the given line. The equation of the given line is $3x+y=3$. To find its slope, we can rearrange this equation into the slope-intercept form, $y=mx+c$, where $m$ is the slope and $c$ is the y-intercept.

Subtract $3x$ from both sides:

Comparing this to $y=mx+c$, we see that the slope of the given line, let's call it $m_1$, is $-3$.

> [!TIP]
> For a linear equation in the form $Ax+By=C$, the slope $m$ can be quickly found using the formula $m = -\frac{A}{B}$. In our case, $A=3$ and $B=1$, so $m_1 = -\frac{3}{1} = -3$.

2. Find the slope of the new line.

The new line is perpendicular to the given line. If $m_1$ is the slope of the given line and $m_2$ is the slope of the new line, then their product must be $-1$.

> [!FORMULA]

> For two perpendicular lines with slopes $m_1$ and $m_2$, we have:

> $m_1 \cdot m_2 = -1$

We found $m_1 = -3$. Now we can find $m_2$:

So, the slope of the new line is $\frac{1}{3}$.

> [!WARNING]
> A common mistake is to forget either the negative sign or the reciprocal when finding the slope of a perpendicular line. Remember it's the *negative reciprocal*.

3. Find the equation of the new line. …