Problem 5 - Precalculus Mathematics for Calculus

Question

Show that the point is on the unit circle. $$\left(\frac{7}{25}, \frac{24}{25}\right)$$

Step-by-Step Solution

Answer

The point \(\left(\frac{7}{25}, \frac{24}{25}\right)\) is on the unit circle.

Step 1: Understand the Condition for a Point to be on the Unit Circle

A point \((x, y)\) lies on the unit circle if it satisfies the equation \(x^2 + y^2 = 1\). This is derived from the definition of the unit circle, which has a radius of 1 and is centered at the origin \((0, 0)\).

Step 2: Substitute the Given Point into the Equation

Substitute \(x = \frac{7}{25}\) and \(y = \frac{24}{25}\) into the unit circle equation: \(x^2 + y^2 = 1\).

Step 3: Calculate \(x^2\) and \(y^2\)

Calculate \(x^2\) by squaring the x-coordinate: \[\left(\frac{7}{25}\right)^2 = \frac{49}{625}\] Calculate \(y^2\) by squaring the y-coordinate: \[\left(\frac{24}{25}\right)^2 = \frac{576}{625}\]

Step 4: Add \(x^2\) and \(y^2\)

Add the results \[\frac{49}{625} + \frac{576}{625} = \frac{625}{625} = 1\]Confirm that this equals 1.

Step 5: Verify and Conclude

Since \(x^2 + y^2 = 1\), the given point \(\left(\frac{7}{25}, \frac{24}{25}\right)\) is on the unit circle.

Key Concepts

Coordinate Geometry

Coordinate geometry is a powerful branch of mathematics that combines algebra with geometry, enabling us to find relationships between geometric shapes using algebraic equations. The most fundamental part of coordinate geometry is the coordinate plane, which consists of an x-axis and a y-axis. Any point in this plane can be described with a pair of numbers

• The first describes its position along the x-axis (horizontal direction)
• The second describes its position along the y-axis (vertical direction)

This pair is often written as (x, y). In coordinate geometry, we use equations to describe different geometric shapes. For example, the equation of a circle in a coordinate plane is derived using its radius and its center. Coordinate geometry helps solve a variety of problems in mathematics by connecting shapes and algebraic expressions.

Equation of a Circle

The equation of a circle is a fundamental concept in coordinate geometry that describes all the points surrounding a central point, called the center, that are at a constant distance, called the radius. For a circle centered at the origin (0, 0), the equation is particularly simple:

• It is expressed as \[x^2 + y^2 = r^2\] where \(r\) is the radius of the circle.

For a unit circle, which has a radius of 1, the equation simplifies to \[x^2 + y^2 = 1\]. This equation represents all the points whose distance from the origin is exactly 1. Any point that satisfies this equation lies on the unit circle. Checking whether a point like \((\frac{7}{25}, \frac{24}{25})\) lies on the unit circle involves verifying whether substituting these values into the unit circle's equation yields 1. In the context of this problem, the point satisfies the unit circle equation, confirming its position on the circle.

Proof Verification

Proof verification is a crucial process in mathematics to ensure that a statement or solution is correct. When we verify a proof, we systematically check each step to ensure there are no errors or assumptions that invalidate the solution. For example, when verifying that a point is on the unit circle, follow the steps:

• First, use the equation of the unit circle \(x^2 + y^2 = 1\).
• Substitute the point's coordinates into the equation.
• Calculate each part separately, like squaring the coordinates.
• Add these values to see if they equal 1.

If all the calculations are correct and the final equation equals 1, the proof that the point is on the unit circle is verified. This meticulous process helps eliminate errors and ensures a solid understanding of the concepts involved.