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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.6.77
Chapter 11, Problem 11.6.77

Theory and Examples


Tangents Find equations for the tangents to the circle (x − 2)² + (y − 1)² = 5 at the points where the circle crosses the coordinate axes.

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Identify the points where the circle crosses the coordinate axes by setting \(y=0\) to find the x-intercepts and \(x=0\) to find the y-intercepts in the circle equation \(\left(x - 2\right)^2 + \left(y - 1\right)^2 = 5\).
Solve for the x-intercepts by substituting \(y=0\) into the equation and solving for \(x\), and solve for the y-intercepts by substituting \(x=0\) and solving for \(y\).
For each intercept point found, calculate the derivative \(\frac{dy}{dx}\) implicitly from the circle equation to find the slope of the tangent line at that point. Use implicit differentiation on \(\left(x - 2\right)^2 + \left(y - 1\right)^2 = 5\).
Use the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point of tangency and \(m\) is the slope found from the derivative, to write the equation of the tangent line at each intercept.
Simplify the tangent line equations to standard or slope-intercept form as needed to express the final equations of the tangents at the points where the circle crosses the coordinate axes.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Equation of a Circle

The equation (x − h)² + (y − k)² = r² represents a circle with center (h, k) and radius r. Understanding this form helps identify the circle's position and size, which is essential for finding points on the circle, such as where it intersects the coordinate axes.
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Parameterizing Equations of Circles & Ellipses

Finding Points of Intersection with Coordinate Axes

To find where a circle crosses the coordinate axes, set x = 0 to find y-intercepts and y = 0 to find x-intercepts. Solving these equations gives the exact points on the circle that lie on the axes, which are needed to determine tangent lines at those points.
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Determining Different Coordinates for the Same Point

Equation of the Tangent Line to a Circle

The tangent line to a circle at a point on its circumference is perpendicular to the radius drawn to that point. Using the slope of the radius, the tangent's slope is the negative reciprocal. The tangent line equation can then be found using point-slope form with the tangent point.
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Related Practice
Textbook Question

Implicitly Defined Parametrizations


Assuming that the equations in Exercises 15−20 define x and y implicitly as differentiable functions x=f(t), y=g(t), find the slope of the curve x=f(t), y=g(t) at the given value of t.


x sin t + 2x = t, t sin t − 2t = y, t = π

Textbook Question

Symmetries and Polar Graphs


Identify the symmetries of the curves in Exercises 1–12. Then sketch the curves in the xy-plane.


r = 1 + 2 sin θ

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Textbook Question

Finding Polar Areas


Find the areas of the regions in Exercises 9–18.


Inside the circle r = 4 sin θ and below the horizontal line r = 3 csc θ

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Textbook Question

Shifting Conic Sections


Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.


9x² + 6y² + 36y = 0

Textbook Question

Finding Cartesian from Parametric Equations


Exercises 1–18 give parametric equations and parameter intervals for the motion of a particle in the xy-plane. Identify the particle’s path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion.


x = 2 sinh t, y = 2 cosh t, −∞<t<∞

Textbook Question

Circles


Sketch the circles in Exercises 53–56. Give polar coordinates for their centers and identify their radii.


r = −2 cos θ

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