Question

Find the value of ‘a’, if the distance between the points $A\left( -3,-14 \right)$ and $B\left( a,-5 \right)$ is 9 units?

Answer 1

Hint: We start solving the problem by drawing the figure representing the given information. We then recall the definition of distance between the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ as $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$. We use this definition for the point $A\left( -3,-14 \right)$ and $B\left( a,-5 \right)$, and equate it to 9 units. We then make the necessary calculation involving squaring, addition and subtractions to get the required value of ‘a’.

Complete step by step answer:
According to the problem, we need to find the value of ‘a’ if the distance between the points $A\left( -3,-14 \right)$ and $B\left( a,-5 \right)$ is 9 units.
Let us draw the figure representing the given information.

We know that the distance between the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$.
So, we have $AB=9$. We use the formula of distance for these points.
So, we have $\sqrt{{{\left( a+3 \right)}^{2}}+{{\left( -5+14 \right)}^{2}}}=9$.
Let us do squaring on both sides.
$\Rightarrow {{\left( \sqrt{{{\left( a+3 \right)}^{2}}+{{\left( 9 \right)}^{2}}} \right)}^{2}}={{\left( 9 \right)}^{2}}$.
$\Rightarrow {{\left( a+3 \right)}^{2}}+81=81$.
$\Rightarrow {{\left( a+3 \right)}^{2}}=0$.
$\Rightarrow a+3=0$.
$\Rightarrow a=-3$.

So, we have found the value of ‘a’ as ‘–3’.

Note: We can see this problem contains a lot of calculation, so we need to perform each step carefully. We can see that the x-coordinates of both the points A and B are the same; this means that the line passing through these points is parallel to the y-axis. We can find the slope and equation of the line after finding the value of ‘a’ which is the x-coordinate of the point B. Similarly, we can expect problems to find the midpoint of the points A and B.