SAT Right triangles Question

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Question 1 · Hard · MathOctober 2024 Batch 5 · Q19Right triangles and trigonometry · 0 solved

$y = x - C$ $y = - 3 (x - 10)^{2}$ In the given system of equations, cis a constant. The system has two distinct real solutions. Which of the following could be the value of c?

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Strategy

Set the two expressions for y equal, then rewrite the result as one equation in x with the parameter isolated. To get two distinct real solutions, analyze when the resulting quadratic has two real roots, which happens when the parameter is large enough to sit above the minimum value of the upward-opening form.

Hints (open one at a time)

Hint 1

Think of each solution as an intersection point between the line and the parabola.

Hint 2

After setting the equations equal, rewrite so the constant is alone on one side and the x-expression is on the other.

Hint 3

The x-expression is an upward-opening quadratic in x; find its minimum value, then choose a constant that is above that minimum so there are two intersections.

Step-by-step Explanation

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