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Holt Algebra Circles Write an equation for a circle. Graph a circle, and identify its center and radius. Objectives circletangent Vocabulary

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Holt Algebra Circles Notes 1. Write an equation for the circle with center (1, –5) and a radius of. 2. Write an equation for the circle with center (–4, 4) and containing the point (–1, 16). 3. Write an equation for the line tangent to the circle x 2 + y 2 = 17 at the point (4, 1). 4. Which points on the graph shown are within 2 units of the point (0, –2.5)?

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Holt Algebra Circles A circle is the set of points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. Because all of the points on a circle are the same distance from the center of the circle, you can use the Distance Formula to find the equation of a circle.

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Holt Algebra Circles Write the equation of a circle with center (–3, 4) and radius r = 6. Example 1: Using the Distance Formula to Write the Equation of a Circle Use the Distance Formula with (x 2, y 2 ) = (x, y), (x 1, y 1 ) = (–3, 4), and distance equal to the radius, 6. Use the Distance Formula. Substitute. Square both sides.

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Holt Algebra Circles Notice that r 2 and the center are visible in the equation of a circle. This leads to the formula for a circle with center (h, k) and radius r. If the center of the circle is at the origin, the equation simplifies to x 2 + y 2 = r 2. Helpful Hint

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Holt Algebra Circles Write the equation of the circle. Example 2A: Writing the Equation of a Circle (x – 0) 2 + (y – 6) 2 = 1 2 x 2 + (y – 6) 2 = 1 the circle with center (0, 6) and radius r = 1 (x – h) 2 + (y – k) 2 = r 2 Equation of a circle Substitute.

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Holt Algebra Circles Use the Distance Formula to find the radius. Substitute the values into the equation of a circle. (x + 4) 2 + (y – 11) 2 = 225 the circle with center (–4, 11) and containing the point (5, –1) (x + 4) 2 + (y – 11) 2 = 15 2 Write the equation of the circle. Example 2B: Writing the Equation of a Circle

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Holt Algebra Circles The location of points in relation to a circle can be described by inequalities. The points inside the circle satisfy the inequality (x – h) 2 + (x – k) 2 r 2.

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Holt Algebra Circles Use the map and information given in Example 3 on page 730. Which homes are within 4 miles of a restaurant located at (–1, 1)? Example 3: Consumer Application The circle has a center (–1, 1) and radius 4. The points insides the circle will satisfy the inequality (x + 1) 2 + (y – 1) 2 < 4 2. Points B, C, D and E are within a 4-mile radius. Point F (–2, –3) is not inside the circle. Check Point F(–2, –3) is near the boundary. (–2 + 1) 2 + (–3 – 1) 2 < 4 2 (–1) 2 + (–4) 2 < < 16 x

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Holt Algebra Circles A tangent is a line in the same plane as the circle that intersects the circle at exactly one point. Recall from geometry that a tangent to a circle is perpendicular to the radius at the point of tangency.

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Holt Algebra Circles Write the equation of the line tangent to the circle x 2 + y 2 = 29 at the point (2, 5). Example 4: Writing the Equation of a Tangent Step 1 Identify the center and radius of the circle. The circle has center of (0, 0) and radius r =. Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula.

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Holt Algebra Circles Example 4 Continued Use the point-slope formula. Rewrite in slope-intercept form. Substitute (2, 5) (x 1, y 1 ) and – for m. 2 5 Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m =. Perpendicular lines have slopes that are opposite reciprocals 2 5 –

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Holt Algebra Circles Example 4 Continued The equation of the line that is tangent to x 2 + y 2 = 29 at (2, 5) is. Graph the circle and the line to check.

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Holt Algebra Circles Notes 1. Write an equation for the circle with center (1, –5) and a radius of. (x – 1) 2 + (y + 5) 2 = Write an equation for the circle with center (–4, 4) and containing the point (–1, 16). (x + 4) 2 + (y – 4) 2 = Write an equation for the line tangent to the circle x 2 + y 2 = 17 at the point (4, 1). y – 1 = –4(x – 4)

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Holt Algebra Circles Notes 4. Which points on the graph shown are within 2 units of the point (0, –2.5)? C, F

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Holt Algebra Circles Circles: Extra Info The following power-point slides contain extra examples and information. Reminder of Lesson Objectives Write an equation for a circle. Graph a circle, and identify its center and radius.

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Holt Algebra Circles Write an equation for a circle. Graph a circle, and identify its center and radius. circletangent Vocabulary

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Holt Algebra Circles Use the Distance Formula to find the radius. Substitute the values into the equation of a circle. (x + 3) 2 + (y – 5) 2 = 169 Find the equation of the circle with center (–3, 5) and containing the point (9, 10). (x + 3) 2 + (y – 5) 2 = 13 2 Check It Out! Example 2

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Holt Algebra Circles Write the equation of the line that is tangent to the circle 25 = (x – 1) 2 + (y + 2) 2, at the point (1, –2). Step 1 Identify the center and radius of the circle. Check It Out! Example 4 From the equation 25 = (x – 1) 2 +(y + 2) 2, the circle has center of (1, –2) and radius r = 5.

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Holt Algebra Circles Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Substitute (5, –5) for (x 2, y 2 ) and (1, –2) for (x 1, y 1 ). Use the slope formula. The slope of the radius is. –3 4 Check It Out! Example 4 Continued Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is.

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Holt Algebra Circles Use the point-slope formula. Rewrite in slope-intercept form. Substitute (5, –5 ) for (x 1, y 1 ) and for m. 4 3 Check It Out! Example 4 Continued Step 3. Find the slope-intercept equation of the tangent by using the point (5, –5) and the slope.

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Holt Algebra Circles Check Graph the circle and the line. Check It Out! Example 4 Continued The equation of the line that is tangent to 25 = (x – 1) 2 + (y + 2) 2 at (5, –5) is.

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