x² – 7xy + 12y² = 0 .

find what straight lines are represented by this equation and determine the angle between them .

To find the straight lines represented by the equation \(x^2 - 7xy + 12y^2 = 0\), we first need to recognize that this is a quadratic equation in two variables, which can be factored to reveal the lines it represents. Let's break it down step-by-step.

Factoring the Equation

The equation can be rewritten in a standard form for factoring. We start with:

x² - 7xy + 12y² = 0

This is a quadratic in \(x\). To factor it, we look for two numbers that multiply to \(12y^2\) (the constant term) and add up to \(-7y\) (the coefficient of the middle term). These two numbers are \(-3y\) and \(-4y\). Thus, we can factor the equation as follows:

• (x - 3y)(x - 4y) = 0

This means the equation represents two straight lines:

• Line 1: \(x = 3y\)
• Line 2: \(x = 4y\)

Finding the Angle Between the Lines

The next step is to determine the angle between these two lines. We can find the slopes of each line by rearranging them into the slope-intercept form \(y = mx + b\), where \(m\) is the slope.

For Line 1 (\(x = 3y\)), we can express it as:

y = (1/3)x

This gives a slope \(m_1 = \frac{1}{3}\).

For Line 2 (\(x = 4y\)), we can rewrite it as:

y = (1/4)x

This gives a slope \(m_2 = \frac{1}{4}\).

Calculating the Angle

The angle \(\theta\) between two lines can be found using the formula:

tan(θ) = |(m1 - m2) / (1 + m1 * m2)|

Substituting the slopes we found:

tan(θ) = |(1/3 - 1/4) / (1 + (1/3 * 1/4))|

Step-by-Step Calculation

• First, find \(m_1 - m_2 = \frac{1}{3} - \frac{1}{4}\).
• Finding a common denominator (which is 12):
• m_1 - m_2 = \frac{4}{12} - \frac{3}{12} = \frac{1}{12}

Next, calculate \(1 + m_1 * m_2\):

• 1 + (1/3 * 1/4) = 1 + \frac{1}{12} = \frac{12}{12} + \frac{1}{12} = \frac{13}{12}

Now we can substitute these values back into the tangent formula:

tan(θ) = |(1/12) / (13/12)| = |1/13|.

Finding the Angle

To find \(\theta\), we take the arctangent:

θ = arctan(1/13)

This angle can be calculated using a calculator or a trigonometric table to find the approximate value. The angle will be relatively small since the slopes are positive and close together.

Summary

The two straight lines represented by the equation \(x^2 - 7xy + 12y^2 = 0\) are \(x = 3y\) and \(x = 4y\). The angle between these lines, calculated through their slopes, is given by \(\theta = arctan(1/13)\), which yields a small angle that indicates they are relatively close to each other in slope.