Dynamic Geometry: P2

Geometry Level 2

The diagram shows a blue parabola $y=ax^{2}+a$ and a black point $P(a;y_{p})$ with $0\le a\le 1$ . As $a$ varies from $0$ to $1$ and back from $1$ to $0$ , the black point moves along a pink curve. The black angle between the two green lines is a right angle. The area bounded by the pink curve and the green lines can be expressed as $\frac{b}{c}$ , where $b$ , and $c$ are coprime positive integers. Find $b+c$ .

The answer is 7.

1 solution

K T
Feb 1, 2021

$P=(a, ax^2+a)$ . Since $a=x$ , the pink curve is given by $y=x^3+x$ , and the area under the pink curve is $A=\int_0^1 (x^3+x) dx = \frac{1}{4}+\frac{1}{2}=\frac{3}{4}$

Dynamic Geometry: P2

The answer is 7.

1 solution

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