Max Min

Algebra Level 3

If x x and y y are real numbers satisfying x 2 + y 2 = 5 x^2+y^2=5 , find the sum of the maximum and minimum possible value of 2 x − y 2x-y .


The answer is 0.

Jun Shin by Jun Shin , 18, Hong Kong

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2 solutions

Jun Shin
Dec 15, 2015

let 2x-y=k, y=2x-k substitute y into x^2+y^2=5 x^2+(2x-k)^2=5 5x^2-4kx+k^2-5=0 x and y are real numbers, so b^2-4ac>=0 b=-4k -4ac=-20k^2+100 16k^2-20k^2+100>=0 -4k^2>=-100 k^2=<25 -5=<k<=5 minimum=-5, maximum=5 therefore, max+min=5-5=0

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Elvin Ding
Jan 1, 2017

View x^2+y^2=5 as a circle. Write 2x-y in standard form: 2x-y=0, y=2x. Substitute 2x into the circle equation. Solve the quadratic, obtain roots of ±1. Substitute +1 and -1 in, and obtain y=±2. Therefore the solutions are (1,2) (-1,-2) Thus the minimum value where y=2x shares a solution with x^2+y^2=5 is (-1,-2) and the maximum value is (1,2). Plug both into 2x-y and you get 0+0=0

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