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Two opposite vertices of a rectangle are $(1,3)$ and $(5,1)$ while the other two vertices lie on the straight line $y=2x+c$. Then the value of $c$ is

- $4$
- $3$
- $-4$
- $-3$

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**Answer $C$**

$\because $ We know that the diagonals of the $RECTANGLE$ bisect each other.

So, equation of the diagonal must be satisfied by the mid-points of the diagonal.

Now, mid-points of diagonal can be calculated by the mid-point formula as:

$\bigg (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\bigg ) =\bigg ( \frac{1+5}{2}, \frac{3+1}{2}\bigg ) = (3, 2)$

The equation of the diagonal is given as:

$$ y = 2x + c$$

Substitute the values of $x$, and $y$ coordinates, we get:

$$2 = 2 \times 3 + c \Rightarrow \color {blue}{c = -4}$$

$\therefore \;C$ is the correct option.