Question

How do you find all values of y such that the distance between $\left( 4,y \right)$ and $\left( -6,4 \right)$ is $24$ ?

Answer 1

Hint: The formula for the distance between two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by the expression $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$ . We put the given points $\left( 4,y \right)$ and $\left( -6,4 \right)$ in this formula and then equate it to the given distance $24$ . We then solve the equation to get the two different values of y.

Complete step-by-step solution:
In coordinate geometry, there are three types of coordinate systems. The first one is the Cartesian coordinate system, the second is the cylindrical coordinate system and the third one is the spherical coordinate system. Out of these three types of coordinate systems, the cartesian coordinate system uses the variables x, y and z. Out of these three variables, the mostly used variables are x and y. The point on the x-y plane is denoted by the notation $\left( x,y \right)$ .
Now, on the x-y plane, the distance between two points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is given by the expression $\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}$ . In our problem, the given points are $\left( 4,y \right)$ and $\left( -6,4 \right)$ . So, putting these points in the above formula, we get,
$\Rightarrow \sqrt{{{\left( 4-\left( -6 \right) \right)}^{2}}+{{\left( y-4 \right)}^{2}}}$
Simplifying the above expression by opening up the brackets, the above expression thus becomes,
$\Rightarrow \sqrt{{{\left( 4+6 \right)}^{2}}+{{\left( y-4 \right)}^{2}}}$
Performing the squares in the above expression, the above expression thus becomes,
$\Rightarrow \sqrt{100+{{\left( y-4 \right)}^{2}}}$
So, we can say that the distance between the two points $\left( 4,y \right)$ and $\left( -6,4 \right)$ is $\sqrt{100+{{\left( y-4 \right)}^{2}}}$ . But it is also given that the distance between the two points is $24$ . So, the expression can be equated to $24$ as,
$\begin{align}
  & \Rightarrow \sqrt{100+{{\left( y-4 \right)}^{2}}}=24 \\
 & \Rightarrow 100+{{\left( y-4 \right)}^{2}}=576 \\
 & \Rightarrow {{\left( y-4 \right)}^{2}}=476 \\
 & \Rightarrow \left( y-4 \right)=\pm 21.81 \\
 & \Rightarrow y=\pm 21.81+4 \\
 & \therefore y=25.81,17.81 \\
\end{align}$
Therefore, we can conclude that the different values of y are $25.81,17.81$ .

Note: For these types of problems, we must remember the formula for the distance between points else we cannot solve them. We should remember to take the two values, positive and negative while performing the square root, else we will get only one value which is wrong.