Question

One vertex of a square $ABCD$ is $A(-1,1)$ and the equation of one diagonal $BD$ is $3x+y-8=0$ then $C$=

• A. $(-5,3)$
• B. $(5,3)$
• C. $(-5,-3)$
• D. $(2,5)$

Answer 1

The correct option is B $(5,3)$

Let point $C(x,y)$.

Slope of $AC=\frac{y-1}{x+1}$

and slope of $BD=3x+y-8=0$

$\begin{array}{c}y=-3x+8\to \left(i\right)\\ \Rightarrow SlopeofBDis-3\\ \Rightarrow AC\perp BD\\ \Rightarrow \frac{y-1}{x+1}\times -3=-1\\ \Rightarrow 3\left(y-1\right)=x+1\\ \Rightarrow 3y-3=x+1\\ \Rightarrow 3y-x=4\to \left(ii\right)\end{array}$

Solving equation (i) and (ii)

$\begin{array}{c}\Rightarrow \frac{x}{20}=\frac{y}{20}=\frac{1}{20}\\ \therefore x=2,y=2\\ \Rightarrow then,byusingmidformula,\frac{x-1}{2}=2,x=5\\ \frac{y+1}{2}=2,y=3\end{array}$