2 Answers

Silk · Answer 1 · 2024-10-21T08:47:47+0000

The sum of all possible values of $a$ is $\frac{8}{5}$.

To find the sum of all possible values of $a$, we first need to find the distance between points $A$ and $B$ using the distance formula.

The distance formula is given by:

$d = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 }$

Using the given coordinates for points $A$ and $B$, we have:

$d = \sqrt{(2 - a)^2 + (2a - 1 + 1)^2}$

Expanding and simplifying the expression, we get:

$d = \sqrt{a^2 - 4a + 4 + 4a^2 - 4a + 1}$

$d = \sqrt{5a^2 - 8a + 5}$

Since the distance between points $A$ and $B$ is given as $\sqrt{5}$, we can equate the expression obtained above to $\sqrt{5}$ and solve for $a$:

$\sqrt{5a^2 - 8a + 5} = \sqrt{5}$

Squaring both sides of the equation, we get:

$5a^2 - 8a + 5 = 5$

Simplifying further, we have:

$5a^2 - 8a = 0$

Factoring out $a$, we get:

$a(5a - 8) = 0$

Setting each factor equal to zero and solving, we find two possible values for $a$: $a = 0$ and $a = \frac{8}{5}$.

Therefore, the sum of all possible values of $a$ is $0 + \frac{8}{5} = \frac{8}{5}$.

Neocotic · Answer 2 · 2024-10-26T06:38:22+0000

${(2a - 1)^(2) - (2 - a)^2} = {5}$ Answer:

0.8

Step-by-step explanation:

$\sqrt{((2a - 1) - (-1))^(2) + (2 - a)^2} = √(5)$ (distance equation)

${(2a)^(2) + (2 - a)^2} = {5}$ (squaring both sides)

${4a^(2) + 4 - 4a + a^2} = {5}$ (expanding the squares)

${5a^(2) - 4a - 1} = {0}$ (Collecting like terms)

Through the quadratic formula:

$a1 = 1\\a2 = -0.2 \\a1 + a2 = 0.8$

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