If The Pair Of Lines Represented By 7 X 2 + 10 X Y − ( P + 1 ) Y 2 = 0 7x^2 + 10xy - (p+1)y^2 = 0 7 X 2 + 10 X Y − ( P + 1 ) Y 2 = 0 Are Perpendicular To Each Other, Find The Value Of P P P .

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Introduction

In mathematics, the concept of perpendicular lines is crucial in understanding various geometric and algebraic properties. When two lines are perpendicular, their slopes are negative reciprocals of each other. In this article, we will explore how to find the value of $p$ in the given quadratic equation, which represents a pair of lines that are perpendicular to each other.

Understanding the Quadratic Equation

The given quadratic equation is $7x^2 + 10xy - (p+1)y^2 = 0$. This equation represents a pair of lines in the form of $ax^2 + 2hxy + by^2 = 0$. To determine the value of $p$, we need to understand the properties of this equation and how it relates to the concept of perpendicular lines.

Condition for Perpendicular Lines

For two lines to be perpendicular, the product of their slopes must be -1. In the context of the given quadratic equation, the slopes of the lines can be found using the coefficients of the equation. The slope of the first line is given by $m_1 = \frac{-10x + 2(p+1)y}{7x}$, and the slope of the second line is given by $m_2 = \frac{10x - 2(p+1)y}{-7x}$.

Deriving the Condition for Perpendicular Lines

To find the value of $p$, we need to derive the condition for perpendicular lines from the given quadratic equation. We can start by finding the product of the slopes of the two lines, which must be equal to -1 for the lines to be perpendicular.

m_1 \cdot m_2 = \frac{(-10x + 2(p+1)y)(10x - 2(p+1)y)}{7x \cdot -7x} = -1

Simplifying the expression, we get:

\frac{(10x - 2(p+1)y)^2}{49x^2} = -1

Cross-multiplying and simplifying further, we get:

(10x - 2(p+1)y)^2 = -49x^2

Expanding the left-hand side of the equation, we get:

100x^2 - 40(p+1)xy + 4(p+1)^2y^2 = -49x^2

Rearranging the terms, we get:

149x^2 - 40(p+1)xy + 4(p+1)^2y^2 = 0

Finding the Value of $p$

To find the value of $p$, we need to compare the derived equation with the original quadratic equation. We can see that the coefficients of the $x^2$ and $y^2$ terms are the same in both equations. Therefore, we can equate the coefficients of the $xy$ term to find the value of $p$.

-40(p+1) = -10

Solving for $p$, we get:

p + 1 = \frac{10}{40} = \frac{1}{4}

Subtracting 1 from both sides, we get:

p = \frac{1}{4} - 1 = -\frac{3}{4}

Conclusion

In this article, we have explored how to find the value of $p$ in the given quadratic equation, which represents a pair of lines that are perpendicular to each other. We have derived the condition for perpendicular lines from the given quadratic equation and solved for the value of $p$. The final answer is $p = -\frac{3}{4}$.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Perpendicular Lines" by Math Is Fun
• [3] "Solving Quadratic Equations" by Khan Academy
  

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Introduction

In our previous article, we explored how to find the value of $p$ in the given quadratic equation, which represents a pair of lines that are perpendicular to each other. In this article, we will answer some frequently asked questions related to the topic.

Q1: What is the condition for two lines to be perpendicular?

A1: For two lines to be perpendicular, the product of their slopes must be -1.

Q2: How do we find the slopes of the lines in the given quadratic equation?

A2: We can find the slopes of the lines by using the coefficients of the equation. The slope of the first line is given by $m_1 = \frac{-10x + 2(p+1)y}{7x}$, and the slope of the second line is given by $m_2 = \frac{10x - 2(p+1)y}{-7x}$.

Q3: How do we derive the condition for perpendicular lines from the given quadratic equation?

A3: We can start by finding the product of the slopes of the two lines, which must be equal to -1 for the lines to be perpendicular. We can then simplify the expression and equate the coefficients of the $xy$ term to find the value of $p$.

Q4: What is the final answer for the value of $p$?

A4: The final answer for the value of $p$ is $p = -\frac{3}{4}$.

Q5: Can we use the same method to find the value of $p$ for other quadratic equations?

A5: Yes, we can use the same method to find the value of $p$ for other quadratic equations that represent a pair of lines that are perpendicular to each other.

Q6: What are some common mistakes to avoid when finding the value of $p$?

A6: Some common mistakes to avoid when finding the value of $p$ include:

• Not equating the coefficients of the $xy$ term correctly
• Not simplifying the expression correctly
• Not checking the final answer for consistency with the original equation

Q7: Can we use technology to find the value of $p$?

A7: Yes, we can use technology such as calculators or computer software to find the value of $p$. However, it is still important to understand the underlying mathematics and to check the final answer for consistency with the original equation.

Q8: What are some real-world applications of finding the value of $p$?

A8: Some real-world applications of finding the value of $p$ include:

• Designing buildings and bridges
• Creating computer graphics and animations
• Modeling population growth and disease spread

Conclusion

In this article, we have answered some frequently asked questions related to finding the value of $p$ in the given quadratic equation. We have also discussed some common mistakes to avoid and some real-world applications of the concept.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Perpendicular Lines" by Math Is Fun
• [3] "Solving Quadratic Equations" by Khan Academy
• [4] "Designing Buildings and Bridges" by American Society of Civil Engineers
• [5] "Creating Computer Graphics and Animations" by ACM SIGGRAPH
• [6] "Modeling Population Growth and Disease Spread" by National Institute of General Medical Sciences