Question

Find the point on \[x\]-axis which is equidistant from \[\left( {2, - 5} \right)\] and \[\left( { - 2,9} \right)\].

Answer 1

Hint: For a point to be equidistant from \[2\] given point it means that the distance between the point and the given point will be equal. And we can equate both the distances since both are equal.

Complete step-by-step answer:
Since the required point is on \[x\]-axis, the \[y\]-coordinate will be equal to \[0\].
Therefore we can assume that the required point is\[\left( {x,{\text{ }}0} \right)\].
Distance between \[2\] points is given by the distance formula.
If the \[2\] points are \[\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[\left( {{x_2},{y_2}} \right)\] then the distance between them is given by \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \].
Here, we can say that \[\left( {x,{\text{ }}0} \right)\] is equidistant to \[\left( {2, - 5} \right)\] and \[\left( { - 2,9} \right)\].
We can equate the distances between them using the distance formula because both are equal. Solving this quadratic equation will be the value of \[x\].
Therefore, we can find the required point by replacing \[x\].
Using the distance formula,
\[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
We can write that \[\left( {{x_1},{y_1}} \right)\] will be \[\left( {2, - 5} \right)\] and \[\left( { - 2,9} \right)\].
Also we can write to equate them,
\[\sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {0 + 5} \right)}^2}} = \sqrt {{{\left( {x + 2} \right)}^2} + {{\left( {0 - 9} \right)}^2}} \]
Squaring on both sides,
\[{\left( {\sqrt {{{\left( {x - 2} \right)}^2} + {{\left( {0 + 5} \right)}^2}} } \right)^2} = {\left( {\sqrt {{{\left( {x + 2} \right)}^2} + {{\left( {0 - 9} \right)}^2}} } \right)^2}\]
We get,
\[{\left( {x - 2} \right)^2} + {\left( 5 \right)^2} = {\left( {x + 2} \right)^2} + {\left( { - 9} \right)^2}\]
Expanding \[{\left( {x - 2} \right)^2}\]and \[{\left( {x + 2} \right)^2}\] on both sides
\[{x^2} - 4x + 4 + 25 = {x^2} + 4x + 4 + 81\]
We can cancelling the equal terms, we get
\[ - 8x = 81 - 25\]
On subtracting, we get
\[ - 8x = 56\]
Divided into it,
$x = - 7$
Therefore, the required point is \[\left( { - 7,0} \right)\].

Note: A point is a basic relationship shown on a graph. Each point is defined by a pair of numbers containing two coordinates. A coordinate is one of a set of numbers used to identify the location of a point on a graph. Each point is identified by both an \[x\] and \[y\] coordinate.
And the coordinate grid has two perpendicular lines, or axes, labelled as number lines. The horizontal axis is known as the \[x\]-axis, while the vertical axis is called the \[y\]-axis. The point where the \[x\]-axis and \[y\]-axis intersect is called the origin. The numbers on a coordinate grid are used to locate points.