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Circles 10-2 Warm Up Lesson Presentation Lesson Quiz Holt Algebra2

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**Find the slope of the line that connects each pair of points.**

Warm Up Find the slope of the line that connects each pair of points. 1 6 1. (5, 7) and (–1, 6) 2. (3, –4) and (–4, 3) –1

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**Find the distance between each pair of points.**

Warm Up Find the distance between each pair of points. 3. (–2, 12) and (6, –3) 17 4. (1, 5) and (4, 1) 5

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**Objectives Write an equation for a circle.**

Graph a circle, and identify its center and radius.

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Vocabulary circle tangent

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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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**Example 1: Using the Distance Formula to Write the Equation of a Circle**

Write the equation of a circle with center (–3, 4) and radius r = 6. Use the Distance Formula with (x2, y2) = (x, y), (x1, y1) = (–3, 4), and distance equal to the radius, 6. Use the Distance Formula. Substitute. Square both sides.

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Check It Out! Example 1 Write the equation of a circle with center (4, 2) and radius r = 7.

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**Notice that r2 and the center are visible in the equation of a circle**

Notice that r2 and the center are visible in the equation of a circle. This leads to a general formula for a circle with center (h, k) and radius r.

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**If the center of the circle is at the origin, the equation simplifies to x2 + y2 = r2.**

Helpful Hint

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**Example 2A: Writing the Equation of a Circle**

Write the equation of the circle. the circle with center (0, 6) and radius r = 1 (x – h)2 + (y – k)2 = r2 Equation of a circle (x – 0)2 + (y – 6)2 = 12 Substitute. x2 + (y – 6)2 = 1

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**Example 2B: Writing the Equation of a Circle**

Write the equation of the circle. the circle with center (–4, 11) and containing the point (5, –1) Use the Distance Formula to find the radius. (x + 4)2 + (y – 11)2 = 152 Substitute the values into the equation of a circle. (x + 4)2 + (y – 11)2 = 225

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Check It Out! Example 2 Find the equation of the circle with center (–3, 5) and containing the point (9, 10).

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**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 + (y – k)2 < r2 The points outside the circle satisfy the inequality (x – h)2 + (y – k)2 > r2

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**Example 3: Consumer Application**

Use the map and information given. Which homes are within 4 miles of a restaurant located at (–1, 1)? 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 < 42. Points B, C, D and E are within a 4-mile radius . Check Point F(–2, –3) is near the boundary. (–2 + 1)2 + (–3 – 1)2 < 42 (–1)2 + (–4)2 < 42 x < 16 Point F (–2, –3) is not inside the circle.

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Check It Out! Example 3 What if…? Which homes are within a 3-mile radius of a restaurant located at (2, –1)?

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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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**Example 4: Writing the Equation of a Tangent**

Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5). Step 1 Identify the center and radius of the circle. From the equation x2 + y2 = 29, the circle has center of (0, 0) and radius r = .

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**Substitute (2, 5) for (x2 , y2 ) and (0, 0) for (x1 , y1 ).**

Example 4 Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula. Substitute (2, 5) for (x2 , y2 ) and (0, 0) for (x1 , y1 ). The slope of the radius is 5 2 Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is 2 5 –

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**Use the point-slope formula.**

Example 4 Continued Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m = 2 5 – Use the point-slope formula. Substitute (2, 5) (x1 , y1 ) and – for m. 2 5 Rewrite in slope-intercept form.

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Example 4 Continued The equation of the line that is tangent to x2 + y2 = 29 at (2, 5) is Check Graph the circle and the line.

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Check It Out! Example 4 Write the equation of the line that is tangent to the circle 25 = (x – 1)2 + (y + 2)2, at the point (5, –5).

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Lesson Quiz: Part I 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).

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Lesson Quiz: Part II 3. Which points on the graph shown are within 2 units of the point (0, –2.5)?

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Lesson Quiz: Part III 4. Write an equation for the line tangent to the circle x2 + y2 = 17 at the point (4, 1).

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