Find the point on the -axis which is equidistant from and .

Understand the Question Any point lying on the -axis has a -coordinate of , so it can be represented as . Being equidistant from two points and means the distances are equal: (or ). We use the distance formula to set up and solve an algebraic equation for . Step 1 · Set Up the Distance Equation Let the required point on the -axis be , and let the given points be and .Since is equidistant from and , we have . Using the distance formula: Step 2 · Solve for Squaring both sides: Expand the squares: Subtract from both sides and simplify: Answer > Common Mistakes Assuming Coordinates Incorrectly: Assuming the point on the -axis is instead of . Points on the -axis always have . Sign Error in Expansion: Expanding as instead of .

Coordinate Geometry | Exercise 7.1

Question 7

Find the point on the $x$ -axis which is equidistant from $(2, -5)$ and $(-2, 9)$ .

Solution

Understand the Question

Any point lying on the $x$ -axis has a $y$ -coordinate of $0$ , so it can be represented as $\text{P}(x, 0)$ .
Being equidistant from two points $\text{A}(2, -5)$ and $\text{B}(-2, 9)$ means the distances are equal: $\text{PA} = \text{PB}$ (or $\text{PA}^2 = \text{PB}^2$ ).
We use the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ to set up and solve an algebraic equation for $x$ .

Step 1 · Set Up the Distance Equation

Let the required point on the $x$ -axis be $\text{P}(x, 0)$ , and let the given points be $\text{A}(2, -5)$ and $\text{B}(-2, 9)$ .Since $\text{P}$ is equidistant from $\text{A}$ and $\text{B}$ , we have $\text{PA} = \text{PB}$ .

Using the distance formula: $\text{PA} = \sqrt{(2 - x)^2 + (-5 - 0)^2}$ $\text{PB} = \sqrt{(-2 - x)^2 + (9 - 0)^2}$

$\sqrt{(2 - x)^2 + (-5)^2} = \sqrt{(-2 - x)^2 + (9)^2}$

Step 2 · Solve for $x$

Squaring both sides: $(2 - x)^2 + (-5)^2 = (-2 - x)^2 + (9)^2$

Expand the squares: $(4 - 4x + x^2) + 25 = (4 + 4x + x^2) + 81$

$x^2 - 4x + 29 = x^2 + 4x + 85$

Subtract $x^2$ from both sides and simplify: $-4x + 29 = 4x + 85$

\begin{aligned} 29 - 85 &= 4x + 4x \\[0.6em] -56 &= 8x \end{aligned}

\begin{aligned} x &= \dfrac{-56}{8} \\[0.6em] x &= -7 \end{aligned}

Answer

$(-7, 0)$

Common Mistakes

Assuming Coordinates Incorrectly: Assuming the point on the $x$ -axis is $(0, y)$ instead of $(x, 0)$ . Points on the $x$ -axis always have $y = 0$ .
Sign Error in Expansion: Expanding $(-2 - x)^2$ as $4 - 4x + x^2$ instead of $[-(2 + x)]^2 = 4 + 4x + x^2$ .

Question 7

More questions in Exercise 7.1