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Chapter 1: Problem 4

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. $$(4,-2) ; 3 x^{2}+y^{2}=52$$

Short Answer

Expert verified
The point (4, -2) lies on the graph of the equation.

Step by step solution

01

Substitute the Point into the Equation

Start by substituting the coordinates of the given point \((4, -2)\) into the equation \(3x^2 + y^2 = 52\). So, \(x=4\) and \(y=-2\) in our equation gives: \[3(4)^2 + (-2)^2 = 52.\]
02

Calculate the Values

Calculate \(3(4)^2\) and \((-2)^2\). First, find \(4^2 = 16\), thus \(3(4)^2 = 3 \times 16 = 48\), and \((-2)^2 = 4\). So, our equation becomes: \(48 + 4\).
03

Check the Equality

Add the calculated values: \(48 + 4 = 52\). Compare this result with the right-hand side of the original equation, which is 52.
04

Conclusion

Since \(48 + 4 = 52\) matches the original equation \(3x^2 + y^2 = 52\), the point \((4, -2)\) satisfies the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation of a Graph
In coordinate geometry, the equation of a graph is an expression that defines a set of points in a coordinate plane. Each point \(x, y\) on a graph satisfies this equation. For example, the equation \(3x^2 + y^2 = 52\) is a type of conic section known as an ellipse. Every point on this ellipse will satisfy the equation when substituted for \(x\) and \(y\).

This form helps us understand the spatial relationship and positioning of points relative to the shape. By defining the equation parameters (in this case using values 3 and 52), we can predict, draw, or confirm whether certain points belong to the graph. This forms the basis for checking if certain coordinates exist on the graph without drawing it. It relies on algebraic manipulation and substitution to verify equations.
Point Substitution
Point substitution is an essential process in verifying if a given point lies on a particular graph. It involves replacing the variables \(x\) and \(y\) in the equation with the respective coordinates of the point. For the point \( (4, -2) \), we substitute \( x = 4 \) and \( y = -2 \) into the equation \(3x^2 + y^2 = 52\).

Here's how substitution works step-by-step:
  • Replace \(x\) with 4: \(3(4)^2 + y^2\)
  • Replace \(y\) with -2: \(3(4)^2 + (-2)^2\)

This substitution transforms the equation into a numeric expression that we can evaluate. By substituting and simplifying, we transform abstract expressions into concrete numbers, making it easy to check for equality and confirm point inclusion on the graph.
Checking Solutions
After substituting the coordinates into the equation, the next step is checking solutions by performing arithmetic operations to verify if the left-hand side equals the right-hand side of the equation. This involves computing the terms separately and then adding them together.

First, calculate \(3(4)^2\):
  • Calculate \(4^2 = 16\)
  • Compute \(3 imes 16 = 48\)
Next, calculate \((-2)^2\):
  • Compute \((-2) imes (-2) = 4\)

Adding these computed values, \(48 + 4\), results in 52. This matches the right-hand side of the original equation \(3x^2 + y^2 = 52\). If these values are equal, as they are in our case, it confirms the point does indeed lie on the graph. Through this method, we ensure the integrity of our graphing and algebraic work, proving the equation's hold over specific coordinates.

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Most popular questions from this chapter

Graph the equation after determining the \(x\) - and \(y\) -intercepts and whether the graph possesses any of the three types of symmetry described on page 58 $$y=-1 / x$$

Rewrite each statement using absolute value notation, as in Example 5. The distance between \(x\) and 1 is at least \(1 / 2\).

(a) Use a graphing utility to graph the equation. (b) Use a graphing utility, as in Example \(5,\) to estimate to one decimal place the \(x\) -intercepts. (c) Use algebra to determine the exact values for the \(x\) -intercepts. Then use a calculator to check that the answers are consistent with the estimates obtained in part (b). $$y=2 x^{2}+x-5$$

The set of real numbers satisfying the given inequality is one or more intervals on the number line. Show the interval(s) on a number line. $$|x-3| \leq 4$$

This exercise outlines a proof of the fact that two nonvertical lines with slopes \(m_{1}\) and \(m_{2}\) are perpendicular if and only if \(m_{1} m_{2}=-1 .\) In the following figure, we've assumed that our two nonvertical lines \(y=m_{1} x\) and \(y=m_{2} x\) intersect at the origin. [If they did not intersect there, we could just as well work with lines parallel to these that do intersect at \((0,0),\) recalling that parallel lines have the same slope.] The proof relies on the following geometric fact: \(\overline{O A} \perp \overline{O B} \quad\) if and only if \(\quad(O A)^{2}+(O B)^{2}=(A B)^{2}\). (a) Verify that the coordinates of \(A\) and \(B\) are \(A\left(1, m_{1}\right)\) and \(B\left(1, m_{2}\right)\) (b) Show that $$\begin{aligned}O A^{2} &=1+m_{1}^{2} \\\O B^{2} &=1+m_{2}^{2} \\\A B^{2} &=m_{1}^{2}-2 m_{1} m_{2}+m_{2}^{2} \end{aligned}$$ (c) Use part (b) to show that the equation $$O A^{2}+O B^{2}=A B^{2}$$ is equivalent to \(m_{1} m_{2}=-1\) (GRAPH CAN'T COPY)
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