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10. Consider a parabola y = x2 and a circle with center C(0,2), as shown below. The points A and B are special points where the parabola and circle are mutually tangential. A 22 -1 I B 2 X (a) Let (x, y) be a point on the parabola, where x 0. Demonstrate the line from (x, y) which also passes through the center of the circle must have a slope of x² - 2 X (b) Let (b, b²) be the coordinates of B, the rightmost point of tangency. A special fact about the parabola y = x² is that its slope at a point (x, y) is always given by the formula m = 2x, so at x=b, the slope is 2b. Use this, along with your result from (a) to determine the value of these coordinates of B. (c) Now determine an equation for the tangent line to the circle at B. Also, sketch a picture of this situation. Show your tangent line, the circle, and parabola all on one graph. (d) The normal line through a point is always perpendicular to the tangent line through that same point. Determine an equation for the normal line to the circle at B. (e) Determine an equation for the circle in terms of x and y. (f) If the line segments AC and BC are drawn, a sector is enclosed by minor arc AB. Show the 7 area of this sector must be equal to -1 Cos The area of a sector, where the central angle 0 is measured in radians, is given by the formula A =

10. Consider a parabola y = x2 and a circle with center C(0,2), as shown below. The points A and B are special points where the parabola and circle are mutually tangential. A 22 -1 I B 2 X (a) Let (x, y) be a point on the parabola, where x 0. Demonstrate the line from (x, y) which also passes through the center of the circle must have a slope of x² - 2 X (b) Let (b, b²) be the coordinates of B, the rightmost point of tangency. A special fact about the parabola y = x² is that its slope at a point (x, y) is always given by the formula m = 2x, so at x=b, the slope is 2b. Use this, along with your result from (a) to determine the value of these coordinates of B. (c) Now determine an equation for the tangent line to the circle at B. Also, sketch a picture of this situation. Show your tangent line, the circle, and parabola all on one graph. (d) The normal line through a point is always perpendicular to the tangent line through that same point. Determine an equation for the normal line to the circle at B. (e) Determine an equation for the circle in terms of x and y. (f) If the line segments AC and BC are drawn, a sector is enclosed by minor arc AB. Show the 7 area of this sector must be equal to -1 Cos The area of a sector, where the central angle 0 is measured in radians, is given by the formula A =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 44E
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10. Consider a parabola y = x² and a circle with center C(0,2), as shown below. The points A and B are special points where the parabola and circle are mutually tangential. -1 3 2 B (a) Let (x, y) be a point on the parabola, where x 0. Demonstrate the line from (x, y) which also passes through the center of the circle must have a slope of x² - 2 COS I (b) Let (b, b²) be the coordinates of B, the rightmost point of tangency. A special fact about the parabola y = x² is that its slope at a point (x, y) is always given by the formula m = 2x, so at x = b, the slope is 2b. Use this, along with your result from (a) to determine the value of these coordinates of B. Now determine an equation for the tangent line to the circle at B. Also, sketch a picture of this situation. Show your tangent line, the circle, and parabola all on one graph. (d) The normal line through a point is always perpendicular to the tangent line through that same point. Determine an equation for the normal line to the circle at B. (e) Determine an equation for the circle in terms of x and y. (f) If the line segments AC and BC are drawn, a sector is enclosed by minor arc AB. Show the area of this sector must be equal to (1/7). The area of a sector, where the central angle 0 is measured in radians, is given by the formula 1 A = ²0.
Transcribed Image Text:10. Consider a parabola y = x² and a circle with center C(0,2), as shown below. The points A and B are special points where the parabola and circle are mutually tangential. -1 3 2 B (a) Let (x, y) be a point on the parabola, where x 0. Demonstrate the line from (x, y) which also passes through the center of the circle must have a slope of x² - 2 COS I (b) Let (b, b²) be the coordinates of B, the rightmost point of tangency. A special fact about the parabola y = x² is that its slope at a point (x, y) is always given by the formula m = 2x, so at x = b, the slope is 2b. Use this, along with your result from (a) to determine the value of these coordinates of B. Now determine an equation for the tangent line to the circle at B. Also, sketch a picture of this situation. Show your tangent line, the circle, and parabola all on one graph. (d) The normal line through a point is always perpendicular to the tangent line through that same point. Determine an equation for the normal line to the circle at B. (e) Determine an equation for the circle in terms of x and y. (f) If the line segments AC and BC are drawn, a sector is enclosed by minor arc AB. Show the area of this sector must be equal to (1/7). The area of a sector, where the central angle 0 is measured in radians, is given by the formula 1 A = ²0.
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