Subject Test Math 1

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Subject Test Math 1

Question 23 of 32

The line with equation $y = 7$ is graphed on the same $x y -plane$ as the circle with center $(4, 5)$ and radius $3$ . What are the $x -coordinates$ 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

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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 $y = 7$ 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 $(4, 5)$ and the line $y = 7$ is the distance between $(4, 5)$ and $(4, 7)$ .)

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 $A B = \sqrt{5}$ . Point $B$ has $x -coordinate$ $4$ . Therefore, the $x -coordinate$ of the points of intersection are $4 - \sqrt{5}$ and $4 + \sqrt{5}$ or $1.76$ and $6.24$ . The correct answer is D.

You could also solve the problem algebraically. A circle with center $(4, 5)$ and radius $3$ has equation ${(x - 4)}^{2} + {(y - 5)}^{2} = 9$ . Substitute $y = 7$ into the equation, and it simplifies to ${(x - 4)}^{2} = 5$ . Solving for $x$ produces the two $x -coordinates$ of the points of intersection of the circle and the line.

Question Difficulty:

hard

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Subject Test Math 1

Question 23 of 32

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