Problem 3 - Precalculus

Question

Determine whether the given point lies on the graph of the equation, as in Example \(1 .\) Note: You are not asked to draw the graph. $$(4,3) ; 3 x^{2}+y^{2}=52$$

Step-by-Step Solution

Answer

The point (4,3) does not lie on the graph of the equation.

Step 1: Identify the point and the equation

We are given the point \((4, 3)\) and the equation \(3x^2 + y^2 = 52\). Our goal is to determine if this point satisfies the equation.

Step 2: Substitute the point into the equation

Substitute \(x = 4\) and \(y = 3\) into the equation \(3x^2 + y^2 = 52\). This gives us \(3(4)^2 + (3)^2\).

Step 3: Calculate each part

Calculate \(3(4)^2\) which is \(3 \times 16 = 48\), and \((3)^2\) which is \(9\).

Step 4: Add the calculated values

Add the values from the previous step: \(48 + 9 = 57\).

Step 5: Compare with the right side of the equation

Compare \(57\) with the right side of the original equation, which is \(52\). Since \(57 eq 52\), the point does not satisfy the equation.

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to use algebraic methods to solve geometric problems. It involves the study of geometric figures using the coordinate plane, which includes an x-axis and a y-axis. In this system:

• Each point on the plane is represented by a pair of numerical values, called coordinates, usually in the form \(x, y\).
• The x-coordinate indicates the position along the horizontal axis, while the y-coordinate indicates the position along the vertical axis.
• The origin is the point where both the x-axis and y-axis intersect, marked as \(0, 0\).
• This system makes it easier to visualize and solve geometric problems using equations.

Coordinate geometry provides a powerful tool for analyzing and interpreting various curves and shapes by expressing them as equations.

Equation of Curves

An equation of a curve represents a set of points that fit a particular mathematical relationship on the coordinate plane. These equations can define lines, circles, parabolas, ellipses, and more complex shapes. For example:

• A linear equation like \(y = mx + b\) represents a straight line.
• A quadratic equation, like \(y = ax^2 + bx + c\), forms a parabola.
• The circle can be described with the equation \(x^2 + y^2 = r^2\), where \(r\) is the radius.

In the given exercise, the equation in question is \(3x^2 + y^2 = 52\), which can be interpreted as an ellipse. However, we are more concerned with checking if a specific point lies on this curve. This involves substituting the point coordinates into the equation and checking if the equation holds true.

Substitution Method

The substitution method is a straightforward technique used to verify if a particular point lies on a given curve or equation. This involves a few simple steps:

• Take the coordinates of the given point, substituting them into the equation one at a time.
• Calculate each term separately to simplify the equation using the substituted values of x and y.
• Add these calculated terms and compare the result to the original equation's right-hand side.

Our previous example followed this exact method. We substituted \(x=4\) and \(y=3\) into \(3x^2 + y^2 = 52\), calculated the individual results as 48 and 9, respectively, summed them to get 57, and saw that it didn't equal 52. Thus, the point \(4, 3\) does not lie on the curve represented by this equation. This simple yet effective method helps in confirming or disproving the presence of a point on a curve.