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 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 6 steps with 6 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
