Example of a Maximum-Minimum Problem (Circle and Lying-Down Parabola)
Author: shiroeExecutions: 10
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| k maximum | |
| k-minimum |
Digit display setting: 6 digits. Digit count can be changed. Real number calculation mode.
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Although this is formally a system of linear equations,
there is, of course, no need to eliminate variables or use trigonometric functions,
so we will not do so.
Geometrically, the point of focus is the intersection of a circle centered at the origin and a horizontally tilted parabola,
so it ultimately boils down to the condition that the radius of the circle is a real number (since the intersection lies on the circumference of the circle).
Let the radius be r.
y^2 = r^2 - x^2
y = ±√(r^2 - x^2)
Since y is a real number, the expression under the square root implies r² - x² ≥ 0 and
x² ≤ r². Using the quadratic formula, we can simplify this to
-r ≤ x ≤ +r. The same applies to y (since the circle has its center at the origin).
In the problem, a = r², and since we assumed a is not negative,
+r is the maximum value of k, and -r is the minimum value of k.
Rewriting x + y² = k as x = ~ and considering it as a quadratic function of y,
k becomes the x-coordinate of the vertex. The vertex lies on the x-axis.
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Although this is formally a system of linear equations,
there is, of course, no need to eliminate variables or use trigonometric functions,
so we will not do so.
Geometrically, the point of focus is the intersection of a circle centered at the origin and a horizontally tilted parabola,
so it ultimately boils down to the condition that the radius of the circle is a real number (since the intersection lies on the circumference of the circle).
Let the radius be r.
y^2 = r^2 - x^2
y = ±√(r^2 - x^2)
Since y is a real number, the expression under the square root implies r² - x² ≥ 0 and
x² ≤ r². Using the quadratic formula, we can simplify this to
-r ≤ x ≤ +r. The same applies to y (since the circle has its center at the origin).
In the problem, a = r², and since we assumed a is not negative,
+r is the maximum value of k, and -r is the minimum value of k.
Rewriting x + y² = k as x = ~ and considering it as a quadratic function of y,
k becomes the x-coordinate of the vertex. The vertex lies on the x-axis.
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find the maximum and minimum values of k for the equation x + y^2 = k.
Assume that a is a positive number not equal to 0.