Problem 99 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find three points that lie on the graph of the equation. (There are many correct answers.) $$x^{2}+y^{2}=49$$

Step-by-Step Solution

Answer

The three points that can lie on the graph of this equation are (7, 0), (0, 7), and approximately (4.95, 4.95).

Step 1: Find the first point

Choose the point where the circle intersects with the x-axis. This happens when \(y = 0\). Solving \(x^{2} + 0^{2} = 49\) leaves us with \(x = \pm \sqrt{49}\). Hence, our first point can be either (7, 0) or (-7, 0).

Step 2: Find the second point

Choose the point where the circle intersects with the y-axis. This happens when \(x = 0\). Solving \(0^{2} + y^{2} = 49\) leaves us with \(y = \pm \sqrt{49}\). Hence, our second point can be either (0, 7) or (0, -7). For our purpose, let's take (0, 7).

Step 3: Find the third point

We'll choose another point on the circle where \(x = y\). Solving \(x^{2} + x^{2} = 49\), we get \(x = \pm \frac{\sqrt{49}}{\sqrt{2}}\). Therefore, the point can be \(\left(\frac{7}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) which simplifies to approximately (4.95, 4.95), or similarly, it can be on the other side of the circle, \(-\left(\frac{7}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)\) approximately (-4.95, -4.95). We consider (4.95, 4.95) for this exercise.

Key Concepts

Circle Equations

In mathematics, circle equations are a fundamental component of coordinate geometry and are utilized to describe the set of all points that maintain a constant distance, known as the radius, from a fixed point called the center. The standard form of the equation of a circle with a center at the origin \( (0,0) \) is given by the equation \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle.

For instance, consider the equation \( x^2 + y^2 = 49 \). This indicates a circle centered at the origin with a radius of \( 7 \) units because \( 49 \) is the square of \( 7 \). Any point \( (x,y) \) that satisfies this equation lies on the circle, and we can find multiple points by choosing different values for \( x \) and \( y \) that make the equation true.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is an area of mathematics that describes geometric shapes and properties using algebraic equations and a coordinate system, usually the Cartesian coordinate system. In this system, any point on a plane is identified by an ordered pair of numbers \( (x,y) \) known as coordinates.

Understanding coordinate geometry requires familiarity with the x and y-axes, which intersect at the origin \( (0,0) \), and the concept of plotting points based on their coordinates. When graphing equations like a circle, we utilize this system to determine the set of points that comply with the equation, such as \( x^{2}+y^{2}=49 \). Through this approach, we can visualize algebraic relationships as geometric shapes on the coordinate plane.

Finding Points on the Graph

In practice, to find points that lie on the graph of a given circle equation, we can select specific values for one variable and solve for the other. For example, setting \( y=0 \) allows us to find where the circle intersects the x-axis, leading to points like (7, 0) and (-7, 0). This illustrates how we can leverage coordinate geometry to solve for specific points and thereby graph the shape described by an equation.

Solving for Intercepts

Solving for intercepts is a common procedure in algebra and coordinate geometry which involves finding the exact points where a graph crosses the x and y-axes. These intercepts provide integral information about the graph's behavior and are particularly significant in the study of linear equations, though they also apply to curves like circles.

For the circle equation \( x^2 + y^2 = 49 \), intercepts are found by setting one variable to zero and solving for the other. When \( y=0 \), we solve for \( x \) to find the x-intercepts, and similarly, setting \( x=0 \) we solve for \( y \) to find the y-intercepts.

Strategies for Finding Intercepts

There can be strategies for selecting points on a circle to cover various aspects of the graph. For instance, to demonstrate a clear understanding of finding intercepts, students could be advised to choose points that not only show where the circle crosses the axis but also examples that portray the circle's symmetry. By exploring intercepts and symmetrical properties together, students gain a more comprehensive grasp of the circle's graphical representation.