Problem 80 Show that the graph of an equati... [FREE SOLUTION]

Chapter 11: Problem 80

Show that the graph of an equation of the form $$A x^{2}+D x+E y+F=0 \quad A \neq 0$$ (a) Is a parabola if $E \neq 0 .$ (b) Is a vertical line if $E=0$ and $D^{2}-4 A F=0$. (c) Is two vertical lines if $E=0$ and $D^{2}-4 A F>0$. (d) Contains no points if $E=0$ and $D^{2}-4 A F<0$.

Short Answer

Expert verified

a) Parabola if $E eq 0$. b) Vertical line if $E = 0$ and $D^{2} - 4 A F = 0$. c) Two vertical lines if $E = 0$ and $D^{2} - 4 A F > 0$. d) No points if $E = 0$ and $D^{2} - 4 A F < 0$.

Step by step solution

Rewrite the given equation

Start with the given equation: \[A x^{2}+ D x + E y + F = 0\]Given: $A eq 0$.

Isolate y if E 鈮� 0

If $ E eq 0 $, isolate $ y $ to identify the conic section: \[ E y = -Ax^{2}- Dx - F \]\[ y = -\frac{A}{E}x^{2} - \frac{D}{E}x - \frac{F}{E} \]This represents a quadratic equation in $ x $, indicating a parabola.

Analyze the equation if E = 0

For parts (b), (c), and (d), analyze the case when $E = 0$:This simplifies the given equation to: \[ A x^{2} + D x + F = 0 \]

Determine the discriminant

Calculate the discriminant of the quadratic equation $ A x^{2} + D x + F = 0 $: \[ \text{Discriminant} = D^{2} - 4 A F \]

Step 5a: Parabola condition: recap

From Step 2, if $ E eq 0 $, the equation represents a parabola.

Step 5b: Vertical line condition

If $ E = 0 $ and $ D^{2} - 4 A F = 0 $:The discriminant is zero, indicating one repeated real root. The quadratic $ A x^{2} + D x + F = 0 $ has a double root, representing a single vertical line.

Step 5c: Two vertical lines condition

If $ E = 0 $ and $ D^{2} - 4 A F > 0 $:The discriminant is positive, indicating two distinct real roots. The quadratic $ A x^{2} + D x + F = 0 $ has two distinct solutions, representing two vertical lines.

Step 5d: No points condition

If $ E = 0 $ and $ D^{2} - 4 A F < 0 $:The discriminant is negative, indicating no real roots. The quadratic $ A x^{2} + D x + F = 0 $ has no real solutions, representing an equation with no points.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

discriminant

The discriminant is a crucial part of understanding quadratic equations. It helps us determine the nature of the roots without solving the equation. Given a quadratic equation in the form: $Ax^2 + Bx + C = 0$The discriminant $ \text{螖} $ is calculated using:$ D^2 - 4AC $

If \text{螖} > 0\, there are two distinct real roots.
If \text{螖} = 0\, there is exactly one real root (a repeated root).
If \text{螖} < 0\, there are no real roots, only complex roots.

In this exercise, the discriminant helps identify the nature of the solutions ensuring if the resulting graph will be a parabola, a vertical line, two vertical lines, or no points at all.

parabola

A parabola is a curve that results from graphing a quadratic equation. It opens either upwards or downwards, and its general form is:y = Ax虏 + Bx + CWhen analyzing the equation $ A x^2 + D x + E y + F = 0 $, if E 鈮� 0, we can isolate y to get:\[ y = -\frac{A}{E}x^2 - \frac{D}{E}x - \frac{F}{E} \]This represents a parabola because it fits the standard quadratic form in x. The coefficient $ -\frac{A}{E} $ dictates whether the parabola opens upwards or downwards.

vertical lines

Vertical lines have the equation x = constant. They appear when the term involving y is absent (i.e., E = 0).

If \ D^2 - 4 A F = 0\, the quadratic equation $Ax^2 + Dx + F = 0$ has one repeated root. This root corresponds to a single vertical line.
If \ D^2 - 4 A F > 0\, the equation has two distinct real roots, meaning the curve intersects the x-axis at two points, represented by two vertical lines.

Therefore, by examining the discriminant when E = 0, we can determine if the graph consists of one or two vertical lines.

real roots

Real roots are the x-values where the quadratic equation equals zero. For the quadratic equation: $Ax^2 + Dx + F = 0$

If \ D^2 - 4 A F = 0\, there is one repeated real root.
If \ D^2 - 4 A F > 0\, there are two distinct real roots.
If \ D^2 - 4 A F < 0\, there are no real roots; instead, the solutions are complex numbers.

In the specific context of this problem, the real roots tell us a lot about the graph's interaction with the x-axis and help classify the resulting curve, completing our understanding of the different graphical scenarios.

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91影视

Show that the graph of an equation of the form $$A x^{2}+D x+E y+F=0 \quad A \neq 0$$ (a) Is a parabola if \(E \neq 0 .\) (b) Is a vertical line if \(E=0\) and \(D^{2}-4 A F=0\). (c) Is two vertical lines if \(E=0\) and \(D^{2}-4 A F>0\). (d) Contains no points if \(E=0\) and \(D^{2}-4 A F<0\).

Short Answer

Step by step solution

Rewrite the given equation

Isolate y if E 鈮� 0

Analyze the equation if E = 0

Determine the discriminant

Step 5a: Parabola condition: recap

Step 5b: Vertical line condition

Step 5c: Two vertical lines condition

Step 5d: No points condition

Key Concepts

discriminant

parabola

vertical lines

real roots

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