1961 AHSME Problems/Problem 40
Problem
Find the minimum value of if .
Solutions
Solution 1
Let , so . Thus, this problem is really finding the shortest distance from the origin to the line .
From the graph, the shortest distance from the origin to the line is the altitude to the hypotenuse of the right triangle with legs and . The hypotenuse is and the area is , so the altitude to the hypotenuse is , which is answer choice .
Solution 2
Solve for in the linear equation. Substitute in . To find the minimum, find the vertex of the quadratic. The x-value of the vertex is . Thus, the minimum value is The answer is .
Solution 3
By Cauchy-Schwarz, Therefore: Therefore: Thus the answer is .
See Also
1961 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 39 |
Followed by Last Question | |
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