Question

The distance between the points \[\left( {a,b} \right)\] and \[\left( { - a, - b} \right)\] is:
(a) \[{a^2} + {b^2}\]
(b) \[\sqrt {{a^2} + {b^2}} \]
(c) 0
(d) \[2\sqrt {{a^2} + {b^2}} \]

Answer 1

Hint:
Here, we will use the formula for distance between two points, and simplify the expression to find the distance between the points \[\left( {a,b} \right)\] and \[\left( { - a, - b} \right)\], and then choose the correct option.
Formula Used: We will use the following formulas:
1) Distance formula: The distance \[d\] between two points \[\left( {{x_1},{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] is given by \[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \].
2) Rule of exponents: If two or more numbers with the same base and different exponents are multiplied, the product can be written as \[{a^b} \times {a^c} = {a^{b + c}}\].

Complete step by step solution:
We will use the distance formula to find the distance between the points \[\left( {a,b} \right)\] and \[\left( { - a, - b} \right)\].
Let \[d\] be the distance between the points \[\left( {a,b} \right)\] and \[\left( { - a, - b} \right)\].
Substituting \[{x_1} = a\], \[{y_1} = b\], \[{x_2} = - a\], and \[{y_2} = - b\] in the distance formula, \[d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \], we get
\[ \Rightarrow d = \sqrt {{{\left( { - a - a} \right)}^2} + {{\left( { - b - b} \right)}^2}} \]
Subtracting the like terms in the parentheses, we get
\[ \Rightarrow d = \sqrt {{{\left( { - 2a} \right)}^2} + {{\left( { - 2b} \right)}^2}} \]
Rewriting the expression \[{\left( { - 2a} \right)^2}\], we get
\[\begin{array}{l} \Rightarrow {\left( { - 2a} \right)^2} = \left( { - 2a} \right) \times \left( { - 2a} \right)\\ \Rightarrow {\left( { - 2a} \right)^2} = \left( { - 1} \right) \times 2{a^1} \times \left( { - 1} \right) \times 2{a^1}\end{array}\]
Rearranging the terms, we get
\[ \Rightarrow {\left( { - 2a} \right)^2} = {\left( { - 1} \right)^2} \times 2 \times 2 \times {a^1} \times {a^1}\]
We know that \[{\left( { - 1} \right)^n}\] is equal to 1 if \[n\] is an even number, and is equal to \[ - 1\] if \[n\] is an odd number.
Thus, we get
\[{\left( { - 1} \right)^2} = 1\]
Therefore, we get
\[ \Rightarrow {\left( { - 2a} \right)^2} = 1 \times 2 \times 2 \times {a^1} \times {a^1}\]
Now by applying rules of exponent, the equation becomes
\[\begin{array}{l} \Rightarrow {\left( { - 2a} \right)^2} = 1 \times 2 \times 2 \times {a^{1 + 1}}\\ \Rightarrow {\left( { - 2a} \right)^2} = 1 \times 2 \times 2 \times {a^2}\end{array}\]
Multiplying the terms of the expression, we get
\[ \Rightarrow {\left( { - 2a} \right)^2} = 4{a^2}\]
Rewriting the expression \[{\left( { - 2b} \right)^2}\], we get
\[\begin{array}{l} \Rightarrow {\left( { - 2b} \right)^2} = \left( { - 2b} \right) \times \left( { - 2b} \right)\\ \Rightarrow {\left( { - 2b} \right)^2} = \left( { - 1} \right) \times 2{b^1} \times \left( { - 1} \right) \times 2{b^1}\end{array}\]
Rearranging the terms, we get
\[ \Rightarrow {\left( { - 2b} \right)^2} = {\left( { - 1} \right)^2} \times 2 \times 2 \times {b^1} \times {b^1}\]
Simplifying the expression, we get
\[ \Rightarrow {\left( { - 2b} \right)^2} = 1 \times 2 \times 2 \times {b^1} \times {b^1}\]
Rewriting using the rule of exponent \[{a^b} \times {a^c} = {a^{b + c}}\], we get
\[\begin{array}{l} \Rightarrow {\left( { - 2b} \right)^2} = 1 \times 2 \times 2 \times {b^{1 + 1}}\\ \Rightarrow {\left( { - 2b} \right)^2} = 1 \times 2 \times 2 \times {b^2}\end{array}\]
Multiplying the terms of the expression, we get
\[ \Rightarrow {\left( { - 2b} \right)^2} = 4{b^2}\]
Now, substituting \[{\left( { - 2a} \right)^2} = 4{a^2}\] and \[{\left( { - 2b} \right)^2} = 4{b^2}\] in the equation \[d = \sqrt {{{\left( { - 2a} \right)}^2} + {{\left( { - 2b} \right)}^2}} \], we get
\[ \Rightarrow d = \sqrt {4{a^2} + 4{b^2}} \]
Factoring out 4 from the terms, we get
\[ \Rightarrow d = \sqrt {4\left( {{a^2} + {b^2}} \right)} \]
Rewriting the expression as a product of square roots, we get
\[ \Rightarrow d = \sqrt 4 \sqrt {{a^2} + {b^2}} \]
\[ \Rightarrow d = 2\sqrt {{a^2} + {b^2}} \]
Therefore, the distance between the points \[\left( {a,b} \right)\] and \[\left( { - a, - b} \right)\] is \[2\sqrt {{a^2} + {b^2}} \].

The correct option is option (d).

Note:
We rewrote \[\sqrt {4\left( {{a^2} + {b^2}} \right)} \] as \[\sqrt 4 \sqrt {{a^2} + {b^2}} \]. This is because if two square roots are multiplied, then the result is equal to the square of the product of the number inside the root. This means that if \[\sqrt x \] and \[\sqrt y \] are multiplied, then the result is equal to \[\sqrt {xy} \].