1. Find the distance between the following pairs of points: (i) $(2, 3)$, $(4, 1)$ (ii) $(-5, 7)$, $(-1, 3)$ (iii) $(a, b)$, $(-a, -b)$

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Step-by-Step Solution

Step 1: Recall the Distance Formula

The distance formula is used to calculate the distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a Cartesian coordinate system. It is derived from the Pythagorean theorem.

Step 2: Calculate Distance for (i) (2, 3) and (4, 1)

For the first pair of points, $(2, 3)$ and $(4, 1)$, we substitute $x_1 = 2$, $y_1 = 3$, $x_2 = 4$, and $y_2 = 1$ into the distance formula. After performing the subtractions and squaring the results, we add them and take the square root. The distance simplifies to $2\sqrt{2}$ units.

Step 3: Calculate Distance for (ii) (-5, 7) and (-1, 3)

For the second pair of points, $(-5, 7)$ and $(-1, 3)$, we substitute $x_1 = -5$, $y_1 = 7$, $x_2 = -1$, and $y_2 = 3$ into the distance formula. We then simplify the expression to find the distance, which is $4\sqrt{2}$ units.

Step 4: Calculate Distance for (iii) (a, b) and (-a, -b)

For the third pair of points, $(a, b)$ and $(-a, -b)$, we substitute $x_1 = a$, $y_1 = b$, $x_2 = -a$, and $y_2 = -b$ into the distance formula. We simplify the algebraic expression by combining like terms, squaring them, and factoring out common terms. The final distance is $2\sqrt{a^2 + b^2}$ units.