Question

Find the values of y for which the distance between the points $P\left( 2,-3 \right)$ and $Q\left( 10,y \right)$ is 10 units?
(a) 8, 2
(b) -9, 3
(c) -9, 5
(d) -8, 2

Answer 1

We start solving the problem by recalling the fact that the distance between two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$. We use this definition to find the distance between the given points P, Q and equate it to 10 units. We then make necessary calculations to get the quadratic equation in y. We then factorize the obtained quadratic equation and equate the factors to zero to get the required values of y.

Complete answer:
According to the problem, we need to find the values of y if the distance between the points $P\left( 2,-3 \right)$ and $Q\left( 10,y \right)$ is 10 units.

We know that the distance between two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$. Now, let us find the distance between the points $P\left( 2,-3 \right)$ and $Q\left( 10,y \right)$ using this result.
So, we get $\sqrt{{{\left( 10-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}}=10$.
$\Rightarrow {{8}^{2}}+{{\left( y+3 \right)}^{2}}={{10}^{2}}$.
$\Rightarrow 64+{{y}^{2}}+6y+9=100$.
$\Rightarrow {{y}^{2}}+6y-27=0$.
Let us now factorize the obtained quadratic equation to find the values of y.
$\Rightarrow {{y}^{2}}+9y-3y-27=0$.
$\Rightarrow y\left( y+9 \right)-3\left( y+9 \right)=0$.
$\Rightarrow \left( y-3 \right)\left( y+9 \right)=0$.
$\Rightarrow y-3=0$, $y+9=0$.
$\Rightarrow y=3$, $y=-9$.
So, we have found the values of y as 3, -9.

$\therefore$ The correct option for the given problem is (b).

Note:
We can also find the values of y from the quadratic equation using the fact that the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$. We should perform each step carefully in order to avoid calculation mistakes and confusion. We can also find the slope of the line passing through the points using the obtained values of y.