Question

The point $(-4,5)$ is the vertex of a square and one of its diagonals is $7x-y+8=0$. The equation of the other diagonal is

• A. $7x–y+23=0$
• B. $7y+x=30$
• C. $7y+x=31$
• D. $x–7y=30$

Answer 1

The correct option is C

$7y+x=31$

Find the equation of the diagonal

Given: The point $(-4,5)$ is the vertex of a square and one of its diagonals is $7x-y+8=0$.

We know that, the diagonals of a square are perpendicular,

So, the required equation of the other diagonal is perpendicular to $7x-y+8=0$

$\Rightarrow y=7x+8$

By comparing with the equation of line in slope-intercept form $y=mx+c$ we get,

The slope of the given line $m_1=7$

We know that if the lines are perpendicular to each other then the product of their slopes is $-1$

So the slope of the required perpendicular line is

$m=-\frac{1}{7}\left[\because 7\times \frac{-1}{7}=-1\right]$

We also know that the equation of line in point slope form is given by: $y-y_1=m\left(x-x_1\right)$

So we have $(x_1,y_1)=(-4,5)$ and $m=\frac{-1}{7}$

By substituting the values we get,

$$\begin{gathered} y-5=\frac{-1}{7}\left(x+4\right) \\ \Rightarrow 7y-35=-x-4 \\ \Rightarrow 7y+x-35+4=0 \\ \Rightarrow 7y+x-31=0 \\ \Rightarrow 7y+x=31 \end{gathered}$$

Therefore, the equation of the other diagonal is $7y+x=31$.

Hence, option (C) is the correct answer.