Given $p \neq Q = 0$, What Is The Equation Of The Line That Passes Through The Points $(-p, -q)$ And $ ( P , Q ) (p, Q) ( P , Q ) [/tex]?A. $y = -x$ B. $y = \frac{q}{p} X$ C. $ Y = Q Y = Q Y = Q [/tex] D.

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Introduction

In mathematics, finding the equation of a line that passes through two given points is a fundamental concept in geometry and algebra. Given two points, we can determine the equation of the line that connects them using the slope-intercept form of a linear equation. In this article, we will explore how to find the equation of a line that passes through the points (−p,−q)(-p, -q) and (p,q)(p, q), where p≠q≠0p \neq q \neq 0.

Understanding the Problem

To find the equation of the line that passes through the points (−p,−q)(-p, -q) and (p,q)(p, q), we need to understand the concept of slope and the slope-intercept form of a linear equation. The slope-intercept form of a linear equation is given by y=mx+by = mx + b, where mm is the slope of the line and bb is the y-intercept.

Finding the Slope

The slope of a line that passes through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by the formula:

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

In this case, we have the points (−p,−q)(-p, -q) and (p,q)(p, q). Plugging these values into the formula, we get:

m=q−(−q)p−(−p)m = \frac{q - (-q)}{p - (-p)}

Simplifying the expression, we get:

m=2q2pm = \frac{2q}{2p}

m=qpm = \frac{q}{p}

Finding the Equation of the Line

Now that we have the slope, we can use the slope-intercept form of a linear equation to find the equation of the line. We know that the line passes through the point (−p,−q)(-p, -q), so we can plug these values into the equation:

y=mx+by = mx + b

−q=qp(−p)+b-q = \frac{q}{p}(-p) + b

Simplifying the expression, we get:

−q=−q+b-q = -q + b

b=0b = 0

Writing the Equation of the Line

Now that we have the slope and the y-intercept, we can write the equation of the line in slope-intercept form:

y=qpx+0y = \frac{q}{p}x + 0

y=qpxy = \frac{q}{p}x

Conclusion

In this article, we explored how to find the equation of a line that passes through the points (−p,−q)(-p, -q) and (p,q)(p, q), where p≠q≠0p \neq q \neq 0. We used the slope-intercept form of a linear equation and the formula for the slope of a line to find the equation of the line. The final answer is:

y=qpxy = \frac{q}{p}x

This equation represents the line that passes through the given points.

Discussion

The equation of the line that passes through the points (−p,−q)(-p, -q) and (p,q)(p, q) is given by y=qpxy = \frac{q}{p}x. This equation represents a line with a slope of qp\frac{q}{p} and a y-intercept of 0. The line passes through the points (−p,−q)(-p, -q) and (p,q)(p, q), and it is a straight line that extends infinitely in both directions.

Example Use Case

Suppose we want to find the equation of the line that passes through the points (−2,−3)(-2, -3) and (2,3)(2, 3). We can plug these values into the equation y=qpxy = \frac{q}{p}x to get:

y=32xy = \frac{3}{2}x

This equation represents the line that passes through the given points.

Final Answer

The final answer is y=qpxy = \frac{q}{p}x.

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Introduction

In our previous article, we explored how to find the equation of a line that passes through two given points. In this article, we will answer some common questions related to finding the equation of a line given two points.

Q: What is the equation of the line that passes through the points (0, 0) and (p, q)?

A: The equation of the line that passes through the points (0, 0) and (p, q) is given by y=qpxy = \frac{q}{p}x. However, if p=0p = 0, then the equation is not defined.

Q: What is the equation of the line that passes through the points (p, q) and (p, -q)?

A: The equation of the line that passes through the points (p, q) and (p, -q) is given by y=qy = q.

Q: What is the equation of the line that passes through the points (p, q) and (-p, q)?

A: The equation of the line that passes through the points (p, q) and (-p, q) is given by x=0x = 0.

Q: What is the equation of the line that passes through the points (p, q) and (p, q)?

A: The equation of the line that passes through the points (p, q) and (p, q) is given by y=qy = q.

Q: How do I find the equation of a line that passes through two points if the slope is not defined?

A: If the slope is not defined, it means that the line is vertical. In this case, the equation of the line is given by x=px = p, where pp is the x-coordinate of one of the points.

Q: Can I find the equation of a line that passes through two points if the points are not on the same plane?

A: No, you cannot find the equation of a line that passes through two points if the points are not on the same plane. This is because the line that passes through two points is a straight line that extends infinitely in both directions, and it cannot be defined if the points are not on the same plane.

Q: How do I find the equation of a line that passes through two points if one of the points is at infinity?

A: If one of the points is at infinity, the equation of the line is given by y=mxy = mx, where mm is the slope of the line.

Q: Can I find the equation of a line that passes through two points if the points are collinear?

A: Yes, you can find the equation of a line that passes through two points if the points are collinear. In this case, the equation of the line is given by y=mx+by = mx + b, where mm is the slope of the line and bb is the y-intercept.

Conclusion

In this article, we answered some common questions related to finding the equation of a line given two points. We hope that this article has been helpful in clarifying any doubts you may have had about finding the equation of a line given two points.

Example Use Cases

  • Finding the equation of a line that passes through two points in a 2D plane.
  • Finding the equation of a line that passes through two points in a 3D space.
  • Finding the equation of a line that passes through two points on a sphere.
  • Finding the equation of a line that passes through two points on a cylinder.

Final Answer

The final answer is that the equation of a line that passes through two points can be found using the slope-intercept form of a linear equation, and the equation is given by y=mx+by = mx + b, where mm is the slope of the line and bb is the y-intercept.