Question

$A(1,3)$ and $C(5,1)$ are two opposite vertices of a rectangle. If the slope of the line on which other two vertices $B$ and $D$ lie is $2$, then equation of $BD$, is

• A. $2x-y=4$
• B. $2x-y=1$
• C. $2x+y=4$
• D. $2x+y=1$

Answer 1

The correct option is A $2x-y=4$

We know that, the diagonals of a rectangle intersects at a point.

Now, here the diagonals are AC and BD.

We know that mid point of any diagonal is the intersection point.

Now, the intersection point is

$(\frac{1+5}{2},\frac{1+3}{2})=(3,2)$

$\Rightarrow (\frac{6}{2},\frac{4}{2})=(3,2)$

We know that any point on the line, is a solution of the line and vice-versa

Hence, $(3,2)$ is a solution of line BD.

Given slope of BD, $m=2$

We know that general form of a line with slope m is $y=mx+c$

Now, taking $y=2;x=3$

$2=2\left(3\right)+C$

$2=6+C$

$\therefore C=2-6=-4$

So, the equation of the line is $y=2x-4$

$\Rightarrow 2x-y=4=0$

$\Rightarrow 2x-y=4$

$\therefore$ So, the answer is A.

$2x-y=4$.