SAT Nonlinear Equations & Systems Question 185 | Free SAT Prep | Aniko

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Question 185·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{aligned} x^2 + y^2 &= r^2 \\ y &= r \end{aligned}

In the system of equations above, $r$ is a positive constant. How many points of intersection do the graphs of the two equations have in the $xy$ -plane?

For systems involving a circle and a line, first recognize the shapes (circle from $x^2 + y^2 = r^2$ , line from $y =$ constant or $y=mx+b$ ). Then use substitution: plug the line’s expression for $y$ into the circle’s equation to get a single equation in $x$ , solve it, and count how many distinct $x$ -values you obtain. Each distinct solution corresponds to an intersection point, which lets you quickly choose the correct answer without fully graphing.

Hints

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Identify the shapes of the graphs

Rewrite each equation in words: $x^2 + y^2 = r^2$ is a circle centered at the origin with radius $r$ , and $y = r$ is a horizontal line. Think about how a horizontal line might meet this circle.

Use substitution

The second equation tells you exactly what $y$ equals. Try substituting $y=r$ into the first equation so everything is in terms of $x$ and $r$ only.

Solve for x and interpret the result

After you substitute, simplify the equation to solve for $x$ . Then think about how many ordered pairs $(x, y)$ you get from this, and what that means for the number of intersection points.

Desmos Guide

Set a positive value for r

In Desmos, type something like r = 3 and make sure $r$ is positive (you can use the slider if you like). Any positive value will work for exploring this system.

Graph the circle

Enter the equation x^2 + y^2 = r^2. You should see a circle centered at the origin with radius $r$ .

Graph the horizontal line

Enter the equation y = r. This will draw a horizontal line at height $r$ .

Count the intersection points

Look at where the line and the circle meet. Use Desmos’s intersection tool (tap where they cross, or use the intersection feature) to see the point(s) of intersection and count how many there are for your chosen positive value of $r$ .

Step-by-step Explanation

Use the second equation to substitute for y

You are given the system:

\begin{aligned} x^2 + y^2 &= r^2 \\ y &= r \end{aligned}

The second equation tells you that in any solution, $y$ must equal $r$ . So in the first equation, you can replace $y$ with $r$ .

Form and simplify an equation in one variable

Substitute $y=r$ into $x^2 + y^2 = r^2$ :

x^2 + r^2 = r^2

Now subtract $r^2$ from both sides:

\begin{aligned} x^2 + r^2 &= r^2 \\ x^2 &= 0 \end{aligned}

Solve this equation for $x$ .

Solve for x and count the intersection points

From $x^2 = 0$ , you get $x = 0$ .

Since $y = r$ , the only ordered pair that satisfies both equations is $(0, r)$ .

Because there is exactly one ordered pair that is a solution to the system, the graphs intersect at one point.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation