Question

Which of the following points is equidistant from (3,-2) and (-5,-2)?
(A) (0, 2)
(B) (0, -2)
(C) (2, 0)
(D) (2, -2)

Answer 1

Hint:
Since we have been told to find the point that is equidistant from our given two points try to consider one option at a time and try to find the distance from the two points to the point in option by using distance formula and hence check which of the distance is obtained equal.

Formula used:
The formula used in this entire sum is Distance formula which is given by $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $

Complete step by step solution:
Let us name point (3, -2) as P and point (-5, -2) as Q.
We know that generally a point is denoted by ${\text{C}}({x_1},{y_1})$ where ${x_1}$and ${y_1}$ are the co-ordinates.
Therefore comparing our given points ${\text{P}}\left( {3, - 2} \right)$ and ${\text{Q}}\left( { - 5, - 2} \right)$ we get
${x_1} = 3,{y_1} = - 2$ and ${x_2} = - 5,{y_2} = - 2$respectively.
Since we have told to find the point equidistant to points ${\text{P}}\left( {3, - 2} \right)$and ${\text{Q}}\left( { - 5, - 2} \right)$, All we have to find is a point say ${\text{R}}$which will have same distance from point ${\text{R}}$to ${\text{P}}\left( {3, - 2} \right)$ and also from ${\text{R}}$to ${\text{Q}}\left( { - 5, - 2} \right)$.
Since we have been given four options we will check the distance between each given point and our points P and Q respectively.
Lets name option (A) (0,2) as point A(0,2)
We will have to find the distance between PA and QA using distance formula and check if they are equal.
Distance formula is given by $\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $
Lets find distance between AP. Here ${x_1} = 3,{x_2} = 0,{y_1} = - 2,{y_2} = 2$
Substituting the values in distance formula we get
$
  {\text{AP = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {0 - 3} \right)}^2} + {{\left( {2 - ( - 2)} \right)}^2}} \\
   = \sqrt {{3^2} + {4^2}} \\
   = \sqrt {9 + 16} \\
   = \sqrt {25} \\
   = 5 \\
 $
Similarly finding distance between QA where ${x_1} = - 5,{x_2} = 0,{y_1} = - 2,{y_2} = 2$
Substituting this values in Distance formula we get
$
  {\text{QA = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {0 - ( - 5)} \right)}^2} + {{\left( {2 - ( - 2)} \right)}^2}} \\
   = \sqrt {{5^2} + {4^2}} \\
   = \sqrt {25 + 16} \\
   = \sqrt {41} \\
 $
Here value of PA is not equal to Value Of QA. So option one is wrong.
Let us consider option B) (0, -2) and name this point B(0, -2)
So we have to find the distance between PB and QB and check if they are equal.
Lets start with finding distance PB where ${x_1} = 3,{x_2} = 0,{y_1} = 2,{y_2} = - 2$
Using distance formula
$
  {\text{PB = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {0 - (3)} \right)}^2} + {{\left( { - 2 - (2)} \right)}^2}} \\
   = \sqrt {{{( - 3)}^2} + {{( - 4)}^2}} \\
   = \sqrt {9 + 16} \\
   = \sqrt {25} \\
   = 5 \\
 $
Now for Distance QB we have ${x_1} = - 5,{x_2} = 0,{y_1} = - 2,{y_2} = - 2$
Using distance formula we get
$
  {\text{QB = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {0 - ( - 5)} \right)}^2} + {{\left( { - 2 - ( - 2)} \right)}^2}} \\
   = \sqrt {{5^2} + 0} \\
   = \sqrt {25} \\
   = 5 \\
 $
Here PB is equal to QB and hence this is the answer to our question.
Further we will check why rest two are not right options.
Consider the third option C)(2,0) and let us name this point as C(2, 0)
Here we have to check if PC and QC have the same distance.
Let Check the distance between PC using distance formula where ${x_1} = 3,{x_2} = 2,{y_1} = 2,{y_2} = 0$
$
  {\text{PC = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {2 - (3)} \right)}^2} + {{\left( {0 - (2)} \right)}^2}} \\
   = \sqrt {{{( - 1)}^2} + {{( - 2)}^2}} \\
   = \sqrt {1 + 4} \\
   = \sqrt 5 \\
 $
Now Lets find distance QC where ${x_1} = - 5,{x_2} = 2,{y_1} = - 2,{y_2} = 0$
$
  {\text{QC = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {2 - ( - 5)} \right)}^2} + {{\left( {0 - ( - 2)} \right)}^2}} \\
   = \sqrt {{7^2} + {2^2}} \\
   = \sqrt {49 + 4} \\
   = \sqrt {53} \\
 $
Here PC and QC are clearly not equal hence C is a wrong option.
Let us consider the fourth option D)(2, -2) and let us name it as D(2, -2)
So we have to find the distance between PD and QD and check if they are same
Let's start with finding the distance between PD where ${x_1} = 3,{x_2} = 2,{y_1} = 2,{y_2} = - 2$
using distance formula we get
$
  {\text{PD = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {2 - (3)} \right)}^2} + {{\left( { - 2 - 2} \right)}^2}} \\
   = \sqrt {{{( - 1)}^2} + {{( - 4)}^2}} \\
   = \sqrt {1 + 16} \\
   = \sqrt {17} \\
 $
Now let us find the distance between QD where ${x_1} = - 5,{x_2} = 2,{y_1} = - 2,{y_2} = - 2$
using distance formula
$
  {\text{QD = }}\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \\
   = \sqrt {{{\left( {2 - ( - 5)} \right)}^2} + {{\left( { - 2 - ( - 2)} \right)}^2}} \\
   = \sqrt {{7^2} + {0^2}} \\
   = \sqrt {49} \\
 $
Hence PD is not equal to QD and thus option D is also wrong.
Hence we can conclude that option B) (0, -2) is the right answer.

Note:
Note that the distance formula used in the entire sum can be used if we have to find the distance between two points. If we have to find the distance between more than two points, the distance formula is very well applicable. We just need to add the number of coordinates according to our requirements.