2Complete the square and write as a squared binomial. 1. x2 + 6x + _____ = _____________2. x2 – 10x + _____ = _____________3. x2 + 24x + _____ = _____________4. x2 + 2x + _____ = _____________5. x2 – 5x + _____ = ______________6.This will be a critical skill we will use later on with conic sections!
3Solve by completing the square: x2 + 6x – 16 = 0x2 – 4x = 11
4Write the equation in standard form for the given circle:
6What we should know already: Chord: A line segment whose endpoints are on the circleSecant: A line that intersects the circle in two pointsRadius: The distance from the center to a point on the circle
7What we should know already: Diameter: A chord that passes through the center of the circleTangent: A line in the plane of a circle that intersects the circle in EXACTLY one point
8Helpful TheoremsIf a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency
9Helpful TheoremsIn a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
10If two lines are Perpendicular… What do we know about their slopes?Their slopes are opposite reciprocals to each other!!!!!!!!Let’s use this to find some tangents….
11Find an equation of the line tangent to the given circle at the given point What’s the slope of the radius to point (-1, 3)?So……..
12Slope of TangentThe slope of the tangent must be the opposite reciprocal of -3, which is
13Writing the Equation Point Slope Form: So to write the equation of our tangent line:
14OR You can use Slope-Intercept form: y = 1/3x + b plug in the point,(-1, 3) to find b
15Find an equation of the line tangent to the given circle at the given point (x – 3)2 + (y + 2)2 = 130at point (-4, 7)
16Write a circular model Cell Phones A cellular phone tower services a 10 mile radius. You get a flat tire 4 miles east and 9 miles north of the tower. Are you in the tower’s range?SOLUTIONIn the diagram above, the origin represents the tower and the positive y-axis represents north.STEP 1Write an inequality for the region covered by the tower. From the diagram, this region is all points that satisfy the following inequality:x2 + y2 < 102
17Write a circular model continued…. STEP 2Substitute the coordinates (4, 9) into the inequality from Step 1.x2 + y2 < 102Inequality from Step 1< 102?Substitute for x and y.97 < 100The inequality is true.ANSWERSo, you are in the tower’s range.