Problem 14 - The Calculus with Analytic Geometry

Question

Find an equation that must be satisfied by the coordinates of any point that is equidistant from the two points \((-3,2)\) and \((4,6)\).

Step-by-Step Solution

Answer

14x + 8y - 39 = 0.

Step 1: Distance Formula

Recall that the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 2: Set up the Distance Equations

Let the coordinates of any point that is equidistant from \((-3,2)\) and \((4,6)\) be \((x,y)\). The distances from \((x,y)\) to \((-3,2)\) and \((4,6)\) must be equal.

Step 3: Equate the Distances

Set the distance from \((x,y)\) to \((-3,2)\) equal to the distance from \((x,y)\) to \((4,6)\): \[\begin{align*} \sqrt{(x + 3)^2 + (y - 2)^2} &= \sqrt{(x - 4)^2 + (y - 6)^2} \end{align*}\]

Step 4: Square Both Sides

Square both sides of the equation to eliminate the square roots: \[\begin{align*} (x + 3)^2 + (y - 2)^2 &= (x - 4)^2 + (y - 6)^2 \end{align*}\]

Step 5: Expand Both Sides

Expand both sides: \[\begin{align*} (x + 3)^2 + (y - 2)^2 &= x^2 + 6x + 9 + y^2 - 4y + 4 \ (x - 4)^2 + (y - 6)^2 &= x^2 - 8x + 16 + y^2 - 12y + 36 \end{align*}\]

Step 6: Simplify and Combine Like Terms

Combine like terms and simplify: \[\begin{align*} x^2 + 6x + 9 + y^2 - 4y + 4 &= x^2 - 8x + 16 + y^2 - 12y + 36 \ x^2 + 6x + 9 + y^2 - 4y + 4 - x^2 + 8x - 16 - y^2 + 12y - 36 &= 0 \ 6x + 8x - 4y + 12y + 9 + 4 - 16 - 36 &= 0 \ 14x + 8y - 39 &= 0 \end{align*}\]

Step 7: Final Equation

Rearrange to find the final equation: \[\begin{align*} 14x + 8y - 39 = 0 \end{align*}\]

Key Concepts

Distance Formula

The distance formula is crucial in coordinate geometry for finding the distance between two points. It is derived from the Pythagorean theorem and is given by:
\ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \. This formula takes the difference between the x-coordinates and the y-coordinates of two points, squares them, sums them up, and takes the square root of the result.

For example, between points \( (x_1, y_1) \) and \( (x_2, y_2) \):

• Calculate \((x_2 - x_1)^2\)


• Calculate \((y_2 - y_1)^2\)


• Sum the two results


• Take the square root of this sum

This method is fundamental in various geometry problems, including those that involve finding equidistant points, such as the problem we are solving.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebraic methods to solve geometric problems.

It involves plotting points, lines, and shapes on a coordinate plane and using equations to represent these objects.

In the given problem, we are dealing with points on a coordinate plane and their distances. The main steps include:

• Identifying the coordinates of points


• Using the distance formula to express distances algebraically


• Equating distances to find equations involving coordinates

For instance, to find a point equidistant from \( (-3,2) \) and \( (4,6) \), we use their coordinates and the distance formula to define an equation for the distances.

Understanding how to navigate and operate within the coordinate plane is essential for solving these types of problems effectively.

Solving Equations

Solving equations is a key component of many mathematical problems, including those involving geometry.

In our problem, we started by setting up two equations using the distance formula. The goal was to find points equidistant from two given points.

The main steps for solving the equation were:

• Setting the distances equal to each other


• Eliminating square roots by squaring both sides


• Expanding and simplifying each side


• Combining like terms to isolate the variable

By following these steps, we derived an equation \( 14x + 8y - 39 = 0 \).

This equation represents all the points \( (x, y) \) that are equidistant from the two original points. Mastering these algebraic techniques is fundamental for solving complex geometric problems.