Two points (a,3) and (5,b) are the opposite vertices of a rectangle.If the other two vertices lie on the line y=2x + c which also passes through the point (a/c,b/c) then the value of c:

A) 2√2 - 2

B) 2√2 - 1

C) 1 - 2√2

D) -1 - 2√2

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We know that the diagonal of the rectangle intercepts at the mid point of each diagonal as well as mid point of triangle.

If the mid point is M (p,q) then using points (a,3) and (5,b);

`p = (a+5)/2`

`q = (3+b)/2`

This point M is on `y = 2x+c.`

`(3+b)/2 = 2(a+5)/2+c`

`3+b = 2(a+5)+2c`

`b-2a = 7+2c-----(1)`

Point `(a/c,b/c)` is also on the line `y = 2x+c.`

`b/c = 2xxa/c+c`

`b-2a = c^2 ----(2)`

`(1) = (2)`

`7+2c = c^2`

`c^2-2c-7 = 0`

Solving the quadratic equation will give you;

`c = (2+-sqrt(4+4xx1xx7))/(2xx1)`

`c = (2+-4sqrt2)/2 = 1+-2sqrt2`

So possible answers for c are;

`c = 1+2sqrt2`

`c = 1-2sqrt2`

** Correct answer is C**.

**Sources:**

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