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Algebra

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Let $a$ and $b$ be real numbers, where $a < b$, and let $A = (a,a^2)$ and $B = (b,b^2)$.  The line $\overline{AB}$ (meaning the unique line that contains the point $A$ and the point $B$) has $x$-intercept $(10,0)$ and $y$-intercept $(0,-2)$.  Find $a$ and $b$.  Express your answer as the ordered pair $(a,b)$.

Dec 9, 2023

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Let the slope of line AB be m. Then the slope-intercept equation of line AB is given by [y = mx + b.]Since line AB passes through (10,0) and (0,−2), we can substitute these points into the equation to get \begin{align*} 0 &= m(10) + b, \ -2 &= m(0) + b. \end{align*}These equations simplify to \begin{align*} m &= -0.2, \ b &= -2. \end{align*}Therefore, the equation of line AB is y=−0.2x−2.

The coordinates of point A are (a,a2), and these coordinates satisfy the equation of the line. Thus, [a^2 = -0.2a - 2.]Rearranging the terms, we get [a^2 + 0.2a + 2 = 0.]

By the quadratic formula, [a = \frac{-0.2 \pm \sqrt{0.2^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} = \frac{-0.2 \pm 0.4 \sqrt{49}}{2} = \frac{-0.2 \pm 2\sqrt{7}}{2} = -0.1 \pm \sqrt{7}.]

Since we are given that a

Dec 9, 2023