To solve for $a$, given that the distance between the points $(a, 3)$ and $(2, a)$ is $\sqrt{5}$, we use the distance formula:

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

Substitute the given points: $\sqrt{(2 - a)^2 + (a - 3)^2} = \sqrt{5}$

Square both sides to eliminate the square root: $(2 - a)^2 + (a - 3)^2 = 5$

Expand both terms: $(2 - a)^2 = 4 - 4a + a^2$ $(a - 3)^2 = a^2 - 6a + 9$

Combine: $4 - 4a + a^2 + a^2 - 6a + 9 = 5$

Simplify: $2a^2 - 10a + 13 = 5$

Subtract 5 from both sides: $2a^2 - 10a + 8 = 0$

Simplify further by dividing through by 2: $a^2 - 5a + 4 = 0$

Factorize the quadratic equation: $(a - 4)(a - 1) = 0$

Solve for $a$: $a = 4 \quad \text{or} \quad a = 1$

Final Answer:

The values of $a$ are $\mathbf{4}$ and $\mathbf{1}$.

Would you like a detailed explanation for any step, or have further questions? 😊

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Here are 5 related questions for practice:

1. How do you find the midpoint between two points?
2. What is the equation of a circle with a given radius and center?
3. Solve for $b$ if $(b, -2)$ is 5 units from $(3, b)$.
4. How do you determine the slope between two points?
5. Explain how to solve a quadratic equation by completing the square.

Tip: Always simplify the equation step-by-step to avoid errors in algebraic manipulations.