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Intermediate Algebra / LEVEL 3 · DIFFICULTY 3/5

Conics, Series and Bounds

Recognize common algebraic structures.

3 stages · 24 practice problems · two 6-question assessment forms

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MINI QUEST IA 3.1Ellipse ParametersRead the lesson
MINI QUEST IA 3.2Finite Geometric SumsRead the lesson
MINI QUEST IA 3.3Bounding Positive ExpressionsRead the lesson
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STAGE IA 3.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Ellipse Parameters

Useful preparation: Factors and Unknown Coefficients

Goal: Understand and apply ellipse parameters.

Before you begin: Factors and Unknown Coefficients

PARENT EQUATION

Unit-circle reference relation: $\displaystyle x^{2} + y^{2} = 1$

A unit circle centered at (0, 0). It is a relation, not a single-valued function $\displaystyle y = f\, \left(x\right)$ .

How does the question’s equation change this graph?

$\displaystyle \frac{\left(x - h\right)^{2}}{a^{2}} + \frac{\left(y - k\right)^{2}}{b^{2}} = 1$ shifts the center to (h, k) and stretches the horizontal and vertical radii to a and b.

Check: Read the radii as square roots of the denominators. Do not confuse $\displaystyle a^{2}$ with a or treat the whole ellipse as one function.

First identify this family. Then read the coefficients and shifts in your actual equation; this reference does not replace the question’s given values.

Understand the idea

A standard ellipse centered at the origin has two semiaxis lengths. Its foci lie on the longer axis, and the squared focal distance is the difference of the squared semiaxis lengths.

\displaystyle \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

\displaystyle c^{2} = a^{2} - b^{2}

when

\displaystyle a \ge b

Choose and carry out a method

Identify the larger denominator as $\displaystyle a^{2}$ and the smaller as $\displaystyle b^{2}$ . Compute $\displaystyle c^{2} = a^{2} - b^{2}$ when the question requests the square of the focal distance.

Check the reasoning

The focus lies inside the ellipse, so $\displaystyle c < a$ . Equal semiaxes produce a circle and focal distance zero.

WORKED EXAMPLE 1

An ellipse has equation $\displaystyle \frac{x^{2}}{36} + \frac{y^{2}}{9} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).

For an ellipse, the focal distance satisfies $\displaystyle c^{2} = a^{2} - b^{2}$ .
The larger denominator is $\displaystyle a^{2} = 36$ ; the smaller is $\displaystyle b^{2} = 9$ .
Thus $\displaystyle c^{2} = 27$ . The foci lie inside the major-axis vertices.

WORKED EXAMPLE 2

An ellipse has equation $\displaystyle \frac{x^{2}}{49} + \frac{y^{2}}{16} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).

For an ellipse, the focal distance satisfies $\displaystyle c^{2} = a^{2} - b^{2}$ .
The larger denominator is $\displaystyle a^{2} = 49$ ; the smaller is $\displaystyle b^{2} = 16$ .
Thus $\displaystyle c^{2} = 33$ . The foci lie inside the major-axis vertices.

Common pitfalls

Possible mix-up: Add the semiaxis squares to find $\displaystyle c^{2}$ .

For an ellipse, use their difference.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

What happens to the foci as the two semiaxes approach equality?

Preview the eight practice prompts

Problem 1
An ellipse has equation $\displaystyle \frac{x^{2}}{81} + \frac{y^{2}}{36} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).
Practise this exact problem →
Problem 2
An ellipse has equation $\displaystyle \frac{x^{2}}{100} + \frac{y^{2}}{49} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).
Practise this exact problem →
Problem 3
An ellipse has equation $\displaystyle \frac{x^{2}}{121} + \frac{y^{2}}{64} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).
Practise this exact problem →
Problem 4
An ellipse has equation $\displaystyle \frac{x^{2}}{144} + \frac{y^{2}}{81} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).
Practise this exact problem →
Problem 5
An ellipse has equation $\displaystyle \frac{x^{2}}{169} + \frac{y^{2}}{100} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).
Practise this exact problem →
Problem 6
An ellipse has equation $\displaystyle \frac{x^{2}}{196} + \frac{y^{2}}{121} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).
Practise this exact problem →
Problem 7New context
An elliptical mirror has semimajor axis 15 cm and semiminor axis 12 cm. What is the square of the distance from its center to either focus?
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Problem 8New context
An elliptical mirror has semimajor axis 16 cm and semiminor axis 13 cm. What is the square of the distance from its center to either focus?
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Open stage IA 3.1 in the student workspace →

STAGE IA 3.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Finite Geometric Sums

Useful preparation: Ellipse Parameters

Goal: Understand and apply finite geometric sums.

Before you begin: Ellipse Parameters

Understand the idea

A geometric sequence multiplies each term by a fixed ratio. Multiplying its sum by that ratio shifts the terms so subtraction cancels every interior term.

\displaystyle S_{n} = \frac{a\, \left(r^{n} - 1\right)}{r - 1}

\displaystyle r \ne 1

Choose and carry out a method

Write S and rS with aligned powers, subtract, then divide by $\displaystyle r - 1$ when $\displaystyle r \ne 1$ . Count the number of terms carefully.

Check the reasoning

The first term uses exponent zero. When $\displaystyle r = 1$ , use n times the common term instead of a formula with a zero denominator.

WORKED EXAMPLE 1

Find the sum of the first 5 terms of 3, 6, 12, … .

Subtract the original sum from twice the sum to cancel all middle terms.
$\displaystyle S = 3$ ( $\displaystyle 1 + 2$ +… $\displaystyle +2^{4}$ )= $\displaystyle 3\, \left(2^{5} - 1\right)$ .
The total is 93; the final term alone is only part of the sum.

WORKED EXAMPLE 2

Find the sum of the first 6 terms of 4, 8, 16, … .

Subtract the original sum from twice the sum to cancel all middle terms.
$\displaystyle S = 4$ ( $\displaystyle 1 + 2$ +… $\displaystyle +2^{5}$ )= $\displaystyle 4\, \left(2^{6} - 1\right)$ .
The total is 252; the final term alone is only part of the sum.

252

Common pitfalls

Possible mix-up: The last exponent equals the number of terms.

Starting with exponent zero makes the last exponent $\displaystyle n - 1$ .

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Derive the sum formula by cancellation rather than memorization.

Preview the eight practice prompts

Problem 1
Find the sum of the first 4 terms of 6, 12, 24, … .
Practise this exact problem →
Problem 2
Find the sum of the first 5 terms of 7, 14, 28, … .
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Problem 3
Find the sum of the first 6 terms of 8, 16, 32, … .
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Problem 4
Find the sum of the first 7 terms of 9, 18, 36, … .
Practise this exact problem →
Problem 5
Find the sum of the first 4 terms of 10, 20, 40, … .
Practise this exact problem →
Problem 6
Find the sum of the first 5 terms of 11, 22, 44, … .
Practise this exact problem →
Problem 7New context
A message chain sends 12 messages in its first round and doubles the number each round. How many messages are sent during the first 6 rounds altogether?
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Problem 8New context
A message chain sends 13 messages in its first round and doubles the number each round. How many messages are sent during the first 7 rounds altogether?
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Open stage IA 3.2 in the student workspace →

STAGE IA 3.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Bounding Positive Expressions

Useful preparation: Finite Geometric Sums

Goal: Understand and apply bounding positive expressions.

Before you begin: Finite Geometric Sums

Understand the idea

For nonnegative x and y, $\displaystyle \left(\sqrt{x} - \sqrt{y}\right)^{2} \ge 0$ rearranges to $\displaystyle x + y \ge 2\, \sqrt{xy}$ . A fixed product therefore gives a lower bound for the sum.

\displaystyle x + y \ge 2\, \sqrt{xy}

, equality when

\displaystyle x = y

Choose and carry out a method

Apply the inequality and identify its equality condition $\displaystyle x = y$ . Show that the allowed values can actually meet that condition before claiming a minimum.

Check the reasoning

A bound alone need not be attainable under extra restrictions. Check positivity and every constraint in the problem.

WORKED EXAMPLE 1

Positive real numbers x and y satisfy $\displaystyle xy = 16$ . Find the minimum possible $\displaystyle x + y$ .

For positive numbers, $\displaystyle x + y \ge 2\, \sqrt{xy}$ , with equality exactly when $\displaystyle x = y$ .
$\displaystyle x + y \ge 2\, \sqrt{16} = 8$ . Choose $\displaystyle x = y = 4$ .
The minimum is attained and equals 8. This is the semiperimeter, not the full perimeter.

WORKED EXAMPLE 2

Positive real numbers x and y satisfy $\displaystyle xy = 25$ . Find the minimum possible $\displaystyle x + y$ .

For positive numbers, $\displaystyle x + y \ge 2\, \sqrt{xy}$ , with equality exactly when $\displaystyle x = y$ .
$\displaystyle x + y \ge 2\, \sqrt{25} = 10$ . Choose $\displaystyle x = y = 5$ .
The minimum is attained and equals 10. This is the semiperimeter, not the full perimeter.

Common pitfalls

Possible mix-up: An inequality bound automatically occurs.

Verify an allowed equality case to establish the minimum.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why does making two positive factors more equal reduce their sum at fixed product?

Preview the eight practice prompts

Problem 1
Positive real numbers x and y satisfy $\displaystyle xy = 49$ . Find the minimum possible $\displaystyle x + y$ .
Practise this exact problem →
Problem 2
Positive real numbers x and y satisfy $\displaystyle xy = 64$ . Find the minimum possible $\displaystyle x + y$ .
Practise this exact problem →
Problem 3
Positive real numbers x and y satisfy $\displaystyle xy = 81$ . Find the minimum possible $\displaystyle x + y$ .
Practise this exact problem →
Problem 4
Positive real numbers x and y satisfy $\displaystyle xy = 100$ . Find the minimum possible $\displaystyle x + y$ .
Practise this exact problem →
Problem 5
Positive real numbers x and y satisfy $\displaystyle xy = 121$ . Find the minimum possible $\displaystyle x + y$ .
Practise this exact problem →
Problem 6
Positive real numbers x and y satisfy $\displaystyle xy = 144$ . Find the minimum possible $\displaystyle x + y$ .
Practise this exact problem →
Problem 7New context
A rectangle has area 169 square units. What is the smallest possible sum of its length and width?
Practise this exact problem →
Problem 8New context
A rectangle has area 196 square units. What is the smallest possible sum of its length and width?
Practise this exact problem →

Open stage IA 3.3 in the student workspace →

Browse another level →

Conics, Series and Bounds

Ellipse Parameters

Understand the idea

Choose and carry out a method

Check the reasoning

An ellipse has equation x236+y29=1\displaystyle \frac{x^{2}}{36} + \frac{y^{2}}{9} = 136x2​+9y2​=1. Find c2\displaystyle c^{2}c2, where its foci are (±c,0).

An ellipse has equation x249+y216=1\displaystyle \frac{x^{2}}{49} + \frac{y^{2}}{16} = 149x2​+16y2​=1. Find c2\displaystyle c^{2}c2, where its foci are (±c,0).

Common pitfalls

Explain it to yourself

Finite Geometric Sums

Understand the idea

Choose and carry out a method

Check the reasoning

Find the sum of the first 5 terms of 3, 6, 12, … .

Find the sum of the first 6 terms of 4, 8, 16, … .

Common pitfalls

Explain it to yourself

Bounding Positive Expressions

Understand the idea

Choose and carry out a method

Check the reasoning

Positive real numbers x and y satisfy xy=16\displaystyle xy = 16xy=16. Find the minimum possible x+y\displaystyle x + yx+y.

Positive real numbers x and y satisfy xy=25\displaystyle xy = 25xy=25. Find the minimum possible x+y\displaystyle x + yx+y.

Common pitfalls

Explain it to yourself

An ellipse has equation $\displaystyle \frac{x^{2}}{36} + \frac{y^{2}}{9} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).

An ellipse has equation $\displaystyle \frac{x^{2}}{49} + \frac{y^{2}}{16} = 1$ . Find $\displaystyle c^{2}$ , where its foci are (±c,0).

Positive real numbers x and y satisfy $\displaystyle xy = 16$ . Find the minimum possible $\displaystyle x + y$ .

Positive real numbers x and y satisfy $\displaystyle xy = 25$ . Find the minimum possible $\displaystyle x + y$ .