Question

The point $A(1,3)$ and $C(5,1)$ are the opposite vertices of rectangle. the equation of line passing through other two vertices and of gradient 2, is

• A. $2x+y=0$
• B. $2x-y-4=0$
• C. $2x-y+4=0$
• D. $2x-y=0$

Answer 1

Answer: (B)

$2x-y-4=0$

Let ABCD is rectangle

Let $A(1,3),B(x_1,y_1),C(5,1)$ $D\left(x_{2,}{y}_2\right)$ be the vertices of of the rectangular

We khow that , diagonals of rectangular bisect each other

Let ,$O$ be the mid point of intersection of diagonal $AC$ and $BD$

$\therefore$ Mid point of $AC=$ Mid point of$BD$

Now, $O(3,2)$ lie on $y=2x+c$

$\therefore 2=2\times 3+c$

$c=-4$

So the value of $c$ is -4

$\left(x_{1,}{y}_1\right)$ lie on $y=2x-4$

$\therefore (x_2,y_2)$ lie on $y=2x-4$

Coordinate of $B=(x_{1,}2x_1+2)$$=(4,2\times 4-4)=(4,4)$

Coordinate of $D=(x_{2,}2x_2-4)=(2,2\times 2-4)=(2,0)$

Thus other two vertices of the rectangle are $(4,4)$ and $(2,0)$ .

$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x_2-x_1)$

$y-4=\frac{0-4}{2-4}(x-4)$

$y-4=2(x-4)$

$y-4=2x-8$

$2x-y-8+4=0$

$2x-y-4=0$

Correct option is (B).