SAT Circles Question 47 | Free SAT Prep | Aniko

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Question 47·Medium·Circles

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A marine research buoy sends a signal to any boat within its circular range. On a coordinate map, the boundary of the signal range is modeled by

x^2+y^2-10x+4y=20.

Which choice is the equation of this circle in standard form?

When a circle is given in expanded form $x^2+y^2+Dx+Ey=F$ , convert to standard form by grouping $x$ -terms and $y$ -terms and completing the square for each variable. Keep track of the constants you add so you adjust the right-hand side correctly, then read $(h,k)$ and $r^2$ directly from $(x-h)^2+(y-k)^2=r^2$ .

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Aim for standard form

Standard form is $(x-h)^2+(y-k)^2=r^2$ . Try completing the square in $x$ and in $y$ .

Work with grouped terms

Rewrite the equation as $(x^2-10x)+(y^2+4y)=20$ before completing the squares.

Remember what completing the square adds

For $x^2+bx$ , you add and subtract $(\tfrac{b}{2})^2$ to make a perfect square trinomial.

Desmos Guide

Graph the given equation

In Desmos, enter the implicit equation x^2+y^2-10x+4y=20 to graph the circle.

Graph each answer choice

Enter each option as its own equation (for example, (x-5)^2+(y+2)^2=49). Desmos will graph each circle.

Match the circles

The correct choice is the one whose graph lies exactly on top of the graph from the original equation (they should be the same circle).

Step-by-step Explanation

Group $x$ -terms and $y$ -terms

Start with

x^2+y^2-10x+4y=20.

Group terms:

\left(x^2-10x\right)+\left(y^2+4y\right)=20.

Complete the square

Complete the square for each group:

$x^2-10x=(x-5)^2-25$
$y^2+4y=(y+2)^2-4$

Substitute:

(x-5)^2-25+(y+2)^2-4=20.

Write in standard form

Add $25$ and $4$ to both sides:

(x-5)^2+(y+2)^2=20+25+4=49.

So the circle’s equation in standard form is $(x-5)^2+(y+2)^2=49$ .

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