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# Coordinates

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The length of the segment between the points (2a, a - 4) and (4,-7) is sqrt(18) units. What is the product of all possible values for a?

Jun 7, 2021

#1
+288
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Substituting the given values into the length formula
$(x_1-x_2)^2 + (y_1-y_2)^2 = \ell^2$ we get
$(2a-4)^2 + ((a-4)-(-7))^2 = (\sqrt{18})^2$.
This can be simplified to $a^2-2a+\frac{7}{5}=0$.
We have $\alpha, \beta$ are the roots of the quadratic equation
$(a-\alpha)(a-\beta)=0$ if and only if $a^2-(\alpha+\beta)a+\alpha\beta=0$.
Therefore, the product of the possible values of $a$ is
$\alpha\beta = \frac{7}{5}$.

Jun 9, 2021