Question

If $\mathrm{PM}$is the perpendicular from $\mathrm{P}\left(2,3\right)$ onto the line $x+y=3$ , then the coordinates of $\mathrm{M}$are

• A. $(2,1)$
• B. $(-1,4)$
• C. $(1,2)$
• D. $(4,–1)$

Answer 1

The correct option is C

$(1,2)$

Explanation for correct option :

Step 1: Apply the coordinate formula

Let the coordinates of $M$ be $\left(a,b\right)$.

The equation of the given line can be rewritten as,

$y=-x+3$

We know that the equation of a straight line is given by,

$\mathit{y}\mathbf{=}\mathit{m}\mathit{x}\mathbf{+}\mathit{c}$, where $m$ is the slope of a line and $c$ is the $y-\text{intercept}$.

After comparison, we find that the slope of the given line is $-1$.

Let the slope of this line be $m_1$. Then,

$m_1=-1\dots \left(1\right)$

Step 2: Finding the slope of the perpendicular line PM

When coordinates of two points of a line are known then we can find its slope by the formula. Let us call the slope here as $m_2$.

${\mathit{m}}_{\mathbf{2}}\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{2}}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}}$

Here $y_2=b,y_1=3,x_2=a,\text{and}{x}_1=2$.

Now, the slope will be

$m_2=\frac{b-3}{a-2}\dots \left(2\right)$

Step 3: Determination of the coordinates of $\mathit{M}$

We know that when two lines are perpendicular then the product of their slopes is equal to $-1$.

$\begin{array}{rcl}\therefore m_1{m}_2& =& -1\\ & \Rightarrow & -1\left(\frac{b-3}{a-2}\right)=-1\\ & \Rightarrow & b-3=a-2\\ & \Rightarrow & b-a=1\dots \left(3\right)\end{array}$

Since the point $M$ lies on the line $x+y=3$, then

$a+b=3\dots \left(4\right)$

Add equation $\left(3\right)\text{and}\left(4\right)$,

$\begin{array}{rcl}\therefore b-a+a+b& =& 1+3\\ & \Rightarrow & 2b=4\\ & \Rightarrow & b=2\end{array}$

From this, we find that $a=1$.

Therefore, the coordinates of M are $(1,2)$.

Hence, the correct option is (C).