Find the distance between the points (a, b) and (a, b). [d = sqrt((a  a)^2 + (b  b)^2.]
Question
Find the distance between the points $(a, b)$ and $(a, b)$. [d = \sqrt{(a  a)^2 + (b  b)^2}.]
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Answer
To find the distance between the points $(a, b)$ and $(a, b)$, you can use the distance formula, which is given by: $$ \text{Distance} = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} $$
Let's assign $(x_1, y_1) = (a, b)$ and $(x_2, y_2) = (a, b)$. Plugging these into the distance formula gives:

For the xcoordinates: $$ (a)  (a) = 2a $$

For the ycoordinates: $$ (b)  (b) = 2b $$
Then, the distance calculation becomes: $$ \sqrt{(2a)^2 + (2b)^2} $$
Simplifying the squares: $$ \sqrt{4a^2 + 4b^2} $$
Factoring out the constant '4' under the square root gives: $$ \sqrt{4(a^2 + b^2)} = 2\sqrt{a^2 + b^2} $$
Thus, the distance between $(a, b)$ and $(a, b)$ is: $$ \boxed{2\sqrt{a^2 + b^2}} $$
This formula provides the final answer, establishing the distance between these two points in a Cartesian plane.
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