Question

Find the distance between the following pair of points $\left( {a,b} \right)$ and $\left( { - a, - b} \right)$ .
A. $4{\left( {{a^2} + {b^2}} \right)^{\dfrac{1}{2}}}$
B. $2\sqrt {{a^2} + {b^2}}$
C. $8{\left( {{a^2} + {b^2}} \right)^{\dfrac{1}{2}}}$
D. $3{\left( {{a^2} + {b^2}} \right)^{\dfrac{1}{2}}}$

Answer 1

Hint : The distance between the two points can be obtained by substituting the given two points in the distance formula. The distance formula is $d = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$ between the two points $(x_1)$ and $(x_2)$

Complete step-by-step answer :
The given points are $A\left( {a,b} \right)$ and $B\left( { - a, - b} \right)$ .
The distance between the two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ is given by the formula,
$D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \cdots \left( 1 \right)$
On comparing the given points with the points in the standard formula, we get
${x_1} = a$ , ${y_1} = b$ , ${x_2} = - a$ and ${y_2} = - b$ .
Substitute the obtained values of ${x_1}$ , ${x_2}$ , ${y_1}$ and ${y_2}$ in the equation (1), we get
$\Rightarrow D = \sqrt {{{\left( { - a - a} \right)}^2} + {{\left( { - b - b} \right)}^2}} \cdots \left( 2 \right)$
On simplifying the equation (2), we get
$\Rightarrow D = \sqrt {{{\left( { - 2a} \right)}^2} + {{\left( { - 2b} \right)}^2}} \cdots \left( 3 \right)$
Squaring the terms in the equation (3), we get
$D = \sqrt {4{a^2} + 4{b^2}} \cdots \left( 4 \right)$
Taking out 4 common from the two terms in the square root in order to simplify equation (4), we get
$\Rightarrow D = \sqrt {4\left( {{a^2} + {b^2}} \right)} \cdots \left( 5 \right)$
The value $\sqrt 4 = 2$ , use it in equation (5), we get
$\Rightarrow D = 2\sqrt {{a^2} + {b^2}}$
Thus, the distance between the two points is $A\left( {a,b} \right)$ and $B\left( { - a, - b} \right)$ is $D = 2\sqrt {{a^2} + {b^2}}$

Note : The important step is to realize that, care should be taken while substituting the value in distance formula, $D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}}$ .
The term with negative sign should be placed carefully in the formula.
For instance, the distance between the points $\left( {1,2} \right)$ and $\left( { - 1, - 2} \right)$ can be obtained by substituting the value in the distance formula as,
$D = \sqrt {{{\left( { - 1 - 1} \right)}^2} + {{\left( { - 2 - 2} \right)}^2}} \\\ D = \sqrt {4 + 16} \\\ D = 2\sqrt 5 \;$
Hence, the distance between the points $\left( {1,2} \right)$ and $\left( { - 1, - 2} \right)$ is $2\sqrt 5$ .