MathLab with Mr. Ahmed

MathLab with Mr. Ahmed is a math learning website by Ahmed Amasha, a mathematics teacher teaching the American curriculum. Explore Algebra 2, Precalculus, Calculus, and SAT and PSAT Math Prep through lessons, focused practice and printable resources.

MathLabwith Mr. Ahmed
Conics, Parametric Equations, and Polar Coordinates

10.1 Conics and Calculus

Focus on standard conic equations, conic tangent slopes, focus and axis parameters.

Free preview Explore the lesson before creating an account.

Play games for this section

Before you solve

Ideas in this lesson

01Standard conic equationsWorked steps · five questions02Conic tangent slopesWorked steps · five questions03Focus and axis parametersWorked steps · five questions
Idea 1 of 3

Standard conic equations

Identify the conic from its standard equation. Ellipse and hyperbola focal distances use different signs.

Worked example 1

For y2=8xy^{2} = 8x, find the focus xx-coordinate.
  1. 4p=84p = 8
  2. p=84=2p = \dfrac{8}{4} = 2

Answer: 22

Worked example 2

For x29+y21=1\dfrac{x^{2}}{9} + \dfrac{y^{2}}{1} = 1, find the positive focus xx-coordinate.
  1. a2=9,b2=1  ⟹  c2=8a^{2} = 9, b^{2} = 1 \implies c^{2} = 8
  2. c=8=8≈2.83c = \sqrt{8} = \sqrt{8} \approx 2.83

Answer: 8≈2.83\sqrt{8} \approx 2.83

Idea 2 of 3

Conic tangent slopes

Implicit differentiation gives tangent slopes without choosing an explicit upper or lower branch.

Worked example 1

On x2+y2=4x^{2} + y^{2} = 4, find the tangent slope at (0,2)(0, 2).
  1. 2x+22x + 2 yy ′=0  ⟹  y′=−xy' = 0 \implies y' = -\dfrac{x}{y}
  2. y′=−02=0y' = -\dfrac{0}{2} = 0

Answer: 00

Worked example 2

On x2+y2=25x^{2} + y^{2} = 25, find the tangent slope at (3,4)(3, 4).
  1. 2x+22x + 2 yy ′=0  ⟹  y′=−xy' = 0 \implies y' = -\dfrac{x}{y}
  2. y′=−34=−34≈−0.75y' = -\dfrac{3}{4} = \dfrac{-3}{4} \approx -0.75

Answer: −34≈−0.75\dfrac{-3}{4} \approx -0.75

Idea 3 of 3

Focus and axis parameters

For y2=4y^{2} = 4 px, pp is the signed focus distance. For an ellipse or hyperbola, compute cc from the semiaxes.

Worked example 1

For y2=8xy^{2} = 8x, find the focus xx-coordinate.
  1. 4p=84p = 8
  2. p=84=2p = \dfrac{8}{4} = 2

Answer: 22

Worked example 2

For x29+y21=1\dfrac{x^{2}}{9} + \dfrac{y^{2}}{1} = 1, find the positive focus xx-coordinate.
  1. a2=9,b2=1  ⟹  c2=8a^{2} = 9, b^{2} = 1 \implies c^{2} = 8
  2. c=8=8≈2.83c = \sqrt{8} = \sqrt{8} \approx 2.83

Answer: 8≈2.83\sqrt{8} \approx 2.83

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §10.1,p10.1, p. 682682. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.