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Subject Test Math 1

Question 23 of 32

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SAT Subject Test

The line with equation y invisible times equals 7 is graphed on the same x y text -plane end text as the circle with center left parenthesis 4 comma 5 right parenthesis and radius 3. What are the x text -coordinates end text of the points of intersection of the line and the circle?

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negative 5 and 5

Correct Answer: 
No

negative 1 and 1

Correct Answer: 
No

1.35 and 6.65

Correct Answer: 
No

1.76 and 6.24

Correct Answer: 
Yes

2 and 6

Correct Answer: 
No

For this question, it may help to draw a diagram.

The figure presents a circle and a horizontal line in the x y plane with the origin labeled O. The number 1 is indicated on both axes. The circle lies entirely above the x-axis and to the right of the y-axis. The center of the circle is labeled “4 comma 5,” and the line is labeled “y equals 7.” The line lies above the center of the circle and intersects the circle at two points.

The line y invisible times equals 7 intersects the circle in 2 points. These two points and the center of the circle form a triangle with two sides of length 3 (since the radius of the circle is 3) and with height 2. (The distance between the point left parenthesis 4 comma 5 right parenthesis and the line y invisible times equals 7 is the distance between left parenthesis 4 comma 5 right parenthesis and left parenthesis 4 comma 7 right parenthesis.)

The figure shows a large triangle with horizontal side A C and two other sides, each of length 3, below A C. A vertical line segment is drawn from point B between vertex A and vertex C to the vertex that is opposite side A C. There is a right angle symbol at point B on either side of the line segment, and the length of the line segment is 2.

Using the Pythagorean Theorem, you can find that A B invisible times equals square root of 5 end root. Point B has x text -coordinate end text 4. Therefore, the x text -coordinate end text of the points of intersection are 4 invisible times minus square root of 5 end root and 4 invisible times plus square root of 5 end root or 1.76 and 6.24. The correct answer is D.

You could also solve the problem algebraically. A circle with center left parenthesis 4 comma 5 right parenthesis and radius 3 has equation  left parenthesis x invisible times minus 4 right parenthesis to the power of 2 end exponent invisible times plus left parenthesis y invisible times minus 5 right parenthesis to the power of 2 end exponent invisible times equals 9.  Substitute y invisible times equals 7 into the equation, and it simplifies to left parenthesis x invisible times minus 4 right parenthesis to the power of 2 end exponent invisible times equals 5. Solving for x produces the two x text -coordinates end text of the points of intersection of the circle and the line.

Question Difficulty: 
hard