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Lines and Distance Problem

Created by

@LovelyBanjo9172

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Questions and Answers

What is the first step to find the distance between the lines $-x + y = 2$ and $4x - 3y = 5$?

Find the intersection of the two lines.

Calculate the angle between the two lines.

Use the distance formula for two parallel lines. (correct)

Convert both equations to slope-intercept form.

Which of the following equations represents a line parallel to $-x + y = 2$?

$-x + y = 3$ (correct)

$4x - 3y = 5$

$x + y = 2$

$x - y = 2$

What characteristic do the lines $4x - 3y = 5$ and $6y - 8x = 1$ share?

They intersect at one point.

They are parallel. (correct)

They are the same line.

They are perpendicular.

Which formula can be used to calculate the distance between two parallel lines of the form $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$?

$rac{|C_2 - C_1|}{ ext{sqrt}(A^2 + B^2)}$ Signup and view all the answers

If $d_1$ is the distance between the lines $-x + y = 2$ and $4x - 3y = 5$, and $d_2$ is the distance between $4x - 3y = 5$ and $6y - 8x = 1$, which of the following statements is true?

Both $d_1$ and $d_2$ are equal. Signup and view all the answers

What is the correct formula to calculate the distance between two parallel lines given by the equations $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$?

$\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$ Signup and view all the answers

What are the slopes of the lines $-x + y = 2$ and $x - y = 2$?

Both have a slope of -1 Signup and view all the answers

Given the lines $4x - 3y = 5$ and $6y - 8x = 1$, what will be their standard forms?

$4x - 3y + 5 = 0$ and $8x - 6y + 1 = 0$ Signup and view all the answers

What determines whether two lines are parallel?

Their slopes must be equal Signup and view all the answers

If two lines are equivalent, what can be said about their distances?

The distance is zero Signup and view all the answers

Study Notes

Lines and Distance

The problem involves two sets of lines:
- First set: $x - y = 2$ and $-x + y = 2$
- Second set: $4x - 3y = 5$ and $6y - 8x = 1$
The problem asks to find the distance between lines within each set:
- $d_1$ is the distance between $x - y = 2$ and $-x + y = 2$.
- $d_2$ is the distance between $4x - 3y = 5$ and $6y - 8x = 1$.
To solve for $d_1$ and $d_2$ you'll need to use a formula to calculate the distance between lines.
Note that the problem description includes a potential typo: It mentions a distance labeled $\alpha$ and a distance labeled $\beta$. These designations are not explicitly defined in the problem, but may refer to the same distances as $d_1$ and $d_2$.

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Description

This quiz focuses on calculating the distance between two sets of lines using the appropriate formulas. Specifically, it covers the lines represented by their equations and requires finding distances labeled as $d_1$ and $d_2$. Additionally, be aware of potential typos regarding distance labels in the problem statement.