Parallelogram on the Coordinate Plane
A parallelogram on the coordinate plane is defined by the following lines:
If the area of the parallelogram is , then can be expressed as , where are coprime positive integers. Find the smallest possible value of .
The answer is 19.
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Lets Start with basic area equation of a parallelogram
Area=height*base;
Now see the line are given as directly parallel pairs
so we can find height(perpendicular distance between lines){y=4x+5,y=4x+3} is 2/17^(1/2)--------(1) Now we got height lets look in to base calculation....
Intersection between lines y=4x+5,y=nx+5 On solving we get (0,5)
Intersection between lines y=4x+3,y=nx+5 On solving we get(2/4-n,20-3n/4-n)
Find distance between these two points...on Calculating we get 68^(1/2)/|4-n| --------------------(2) By multiplying 1,2 we get area of parallelogram which is equal to 16(given)
By equating both and on solving we get n=15/4 or 17/4