Question

A point \(P\) is equidistant from (-5,1) and \((4,-4) .\) Find the coordinates of \(P\) if its \(y\) -coordinate is twice its \(x\) -coordinate.

Short answer

The coordinates of point P are (-3, -6).

Step-by-step solution

Understand the Problem

Given a point \(P(x, y)\) that is equidistant from points \((-5,1)\) and \((4,-4)\). Also, given that the \(y\)-coordinate is twice the \(x\)-coordinate, meaning \(y = 2x\).

Use the Distance Formula

The distance formula is: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]. Since \(P\) is equidistant from both given points, set the distances equal to each other:

Set Up the Equations

The distance from \(P(x, 2x)\) to \((-5, 1)\): \[ d_1 = \sqrt{(x + 5)^2 + (2x - 1)^2} \]. The distance from \(P(x, 2x)\) to \((4, -4)\):\[ d_2 = \sqrt{(x - 4)^2 + (2x + 4)^2} \]. Since the distances are equal, set them equal to each other: \[ \sqrt{(x + 5)^2 + (2x - 1)^2} = \sqrt{(x - 4)^2 + (2x + 4)^2} \]

Square Both Sides to Eliminate the Square Roots

\[ (x + 5)^2 + (2x - 1)^2 = (x - 4)^2 + (2x + 4)^2 \]

Expand and Simplify

Expand the terms: \[ (x + 5)^2 = x^2 + 10x + 25 \] \[ (2x - 1)^2 = 4x^2 - 4x + 1 \] \[ (x - 4)^2 = x^2 - 8x + 16 \] \[ (2x + 4)^2 = 4x^2 + 16x + 16 \]Combine them: \[ x^2 + 10x + 25 + 4x^2 - 4x + 1 = x^2 - 8x + 16 + 4x^2 + 16x + 16 \]

Combine Like Terms

\[ 5x^2 + 6x + 26 = 5x^2 + 8x + 32 \]Subtract \(5x^2\) from both sides:\[ 6x + 26 = 8x + 32 \]Subtract \(6x\) and \(32\) from both sides:\[ -6 = 2x \]So, we get:\[ x = -3 \]

Find the y-coordinate

Since \( y = 2x \): \[ y = 2(-3) = -6 \]

State the Coordinates of Point P

Thus, the coordinates of \(P\) are \((-3, -6)\).

Key concepts

Distance Formula

Coordinate geometry often involves finding distances between points on a plane. The distance formula is crucial for this. It is derived from the Pythagorean theorem and is used to calculate the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a coordinate system. The formula is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula helps us understand the direct distance between two points, regardless of whether the points are diagonal, vertical, or horizontal in relation to each other.

• For example, if you have points \( (-5, 1) \) and \( (4, -4) \), the distance calculation would involve substituting these values into the formula.
• This formula can help us determine whether or not a point is at the same distance from two other points, which is key for many geometric problems.

Understanding and using the distance formula is a fundamental skill in coordinate geometry.

Equidistant Points

In geometry, equidistant points are points that are the same distance from a given point or a set of points. For example, if we are given two points, \( (-5,1) \) and \( (4,-4) \), a point \( P(x,y) \) being equidistant from these two points means:
\[ \sqrt{(x + 5)^2 + (2x - 1)^2} = \sqrt{(x - 4)^2 + (2x + 4)^2} \]

• This equation shows that the distances from \( P \) to each of the given points must be equal.
• To find this point \( P \), we can use the distance formula and solve the algebraic equation that results from setting the two distances equal.

This is a common scenario in many math problems where symmetry and balance need to be maintained.

Coordinate System

A coordinate system allows us to specify points on a plane using pairs of numbers, known as coordinates. These coordinates are written as \( (x, y) \). The first number \( x \) denotes the position on the horizontal axis, and the second number \( y \) marks the position on the vertical axis.

• For example, in the problem, the coordinates \( (-5,1) \) and \( (4,-4) \) are used to place points on the Cartesian plane.
• A point P with coordinates \( (x, y) \) can be plotted by moving \( x \) units horizontally and \( y \) units vertically from the origin.

The coordinate system is fundamental for visualizing and solving geometric problems.

Algebraic Equations

Algebraic equations are mathematical statements that use algebraic expressions, containing variables and constants, to depict relationships. In this specific exercise, understanding algebraic equations helps us find the unknown coordinates of point \( P \).

• Given \( y = 2x \), we substitute this into the distance formula. This substitution transforms the geometric problem into an algebraic equation that can be solved step-by-step.
• The key steps include setting the distances equal, squaring both sides to remove the square roots, expanding the terms, and simplifying the equation to isolate and solve for the variable \( x \).

The final result gives us the coordinates of point \( P \). This problem showcases the need to use algebraic manipulation and substitution to find solutions.