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The distance between the points $A$ and $B$ is $\sqrt{5}.$ If $A = (a,-1)$ and $B = (2,2a-7),$ then find the sum of all possible values of $a.$

 Oct 28, 2023
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avatar+624 
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Let's start by plugging the coordinates ofAandBinto the distance formula:

d(A, B) = \sqrt{(2-a)^2 + (-1-2a-7)^2}

We know thatd(A,B)=5​, so we can substitute this into the equation to get:

\sqrt{(2-a)^2 + (-1-2a-7)^2} = \sqrt{5}

Squaring both sides of the equation, we get:

(2-a)^2 + (-1-2a-7)^2 = 5

Expanding the parentheses, we get:

4-4a+a^2 + 1+4a+49 = 5

Simplifying, we get:

a^2 - 4a + 50 = 0

Factoring the quadratic, we get:

(a-10)(a-5) = 0

Therefore,a=10ora=5.

Since we are asked for the sum of all possible values ofa, the sum is 10+5=15​.

 Oct 28, 2023

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