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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 2, 2023
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Since the x-intercept of the line is (10,0), we know that the equation of the line is of the form y=mx+n for some value of m and n. Since the y-intercept of the line is (0,−2), we have that −2=m⋅0+n. Hence, n=−2. Therefore, the equation of the line is y=mx−2.

 

We know that the point (a,a2) lies on the line, so a2=ma−2. We also know that the point (b,b2) lies on the line, so b2=mb−2. Subtracting these two equations, we get b2−a2=m(b−a). Since a 0, so m=b−ab2−a2​.

 

Since the line passes through the point (10,0), we have that 0=m(10)−2. Substituting the expression for m into this equation, we get 0=b−ab2−a2​(10)−2. This simplifies to b2−a2=5a−5b. We also have the equation a2=ma−2. Substituting m=b−ab2−a2​, we get a2=b−ab2−a2​(a)−2. This simplifies to (b−a)a2−(b2−a2)a+2(b−a)=0. This factors as (a−2)(b−a)a2=0.

 

Since a

Substituting a=0 into the equation b2−a2=5a−5b, we get b2=5b. This factors as b2−5b=0, which factors as b(b−5)=0. Therefore, b=0 or b=5. Since a

 

Therefore, (a,b)=(0,5)​.

 Dec 2, 2023

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