4.4 Q-2

Question Statement

Find the equation of the line through:

1. The point $(2,-9)$ and the intersection of the lines $2x + 5y - 8 = 0$ and $3x - 4y - 6 = 0$.
2. The intersection of the lines $x - y - 4 = 0$ and $7x + y + 20 = 0$, which is:
   • Parallel to the line $6x + y - 14 = 0$.
   • Perpendicular to the line $6x + y - 14 = 0$.
3. Through the intersection of the lines $x + 2y + 3 = 0$ and $3x + 4y + 7 = 0$, and making intercepts on the axes.

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Background and Explanation

To solve these problems, we need to:

1. Find the point of intersection of two lines using substitution or elimination.
2. Use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point and $m$ is the slope.
3. Calculate the slope of the line through two points using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
4. Solve for the slope condition if the line is parallel or perpendicular to another line.

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Solution

Part 1: Equation of the Line through $(2, -9)$ and the Intersection of $2x + 5y - 8 = 0$ and $3x - 4y - 6 = 0$

Step 1: Find the point of intersection of the two lines.

We solve the system of equations:

$$2x + 5y - 8 = 0 \tag{1}$$ $$3x - 4y - 6 = 0 \tag{2}$$

Using the elimination method, we manipulate the equations to find the values of $x$ and $y$:

$$\frac{x}{-30 - 32} = \frac{-y}{-12 - 124} = \frac{1}{-8 - 15} \tag{C}$$ $$\frac{x}{-62} = \frac{y}{-12} = \frac{1}{-23} \tag{2}$$

Solving for $x$ and $y$ gives:

$$x = \frac{62}{23}, \quad y = \frac{12}{23}$$

Thus, the point of intersection is $\left[\frac{62}{23}, \frac{12}{23}\right]$.

Step 2: Find the slope of the line through $(2, -9)$ and the point $\left[\frac{62}{23}, \frac{12}{23}\right]$.

The slope formula is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Substituting the values:

$$m = \frac{\frac{12}{23} - (-9)}{\frac{62}{23} - 2} = \frac{12 + 207}{62 - 46} = \frac{219}{16}$$

Step 3: Write the equation of the line.

Using the point-slope form:

$$y - (-9) = \frac{219}{16}(x - 2)$$

Simplifying:

$$16(y + 9) = 219x - 428$$ $$219x - 16y - 438 - 144 = 0$$ $$219x - 16y - 582 = 0$$

This is the required equation of the line.

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Part 2: Equation of the Line Parallel and Perpendicular to $6x + y - 14 = 0$ through the Intersection of $x - y - 4 = 0$ and $7x + y + 20 = 0$

Step 1: Find the general equation of the line through the intersection.

The general form of the line through the intersection of the lines $x - y - 4 = 0$ and $7x + y + 20 = 0$ is:

$$(x - y - 4) + k(7x + y + 20) = 0$$

Simplifying:

$$(1 + 7k)x + (k - 1)y - 4 - 20k = 0 \tag{1}$$

Step 2: Condition for Parallel Line.

For the line to be parallel to $6x + y - 14 = 0$, the slope of the line must be equal to $-6$. The slope of line (1) is:

$$m = -\frac{7k + 1}{k - 1}$$

Equating to $-6$:

$$\frac{7k + 1}{k - 1} = -6$$

Solving for $k$:

$$7k + 1 = -6(k - 1)$$ $$7k + 1 = -6k + 6$$ $$13k = 5$$ $$k = -7$$

Substitute $k = -7$ into equation (1) to get the equation:

$$(1 - 49)x + (-8)y - 4 + 140 = 0$$

Simplifying:

$$-48x - 8y - 144 = 0$$

Dividing by $-1$:

$$6x + y + 18 = 0$$

Step 3: Condition for Perpendicular Line.

For the line to be perpendicular to $6x + y - 14 = 0$, the product of their slopes must be $-1$. The slope of $6x + y - 14 = 0$ is $-6$, so we set up the equation:

$$\frac{7k + 1}{k - 1} \times (-6) = -1$$

Solving for $k$:

$$6(7k + 1) = -(k - 1)$$ $$42k + 6 = -k + 1$$ $$43k = -5$$ $$k = -\frac{5}{43}$$

Substituting $k = -\frac{5}{43}$ into equation (1) gives:

$$\left(\frac{43 - 35}{43}\right)x + \left(\frac{-5 - 43}{43}\right)y - 4 + 20\left(-\frac{5}{43}\right) = 0$$

Simplifying:

$$\frac{8}{43}x - \frac{48}{43}y - 4 = 0$$

Multiplying through by 43:

$$8x - 48y = 172$$

Simplifying:

$$x - 6y - 34 = 0$$

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Part 3: Equation of the Line through the Intersection of $x + 2y + 3 = 0$ and $3x + 4y + 7 = 0$

Step 1: General form of the line through the intersection.

The equation of the line through the intersection of the lines is:

$$(1 + 3k)x + (2 + 4k)y + 3 + 7k = 0$$

Step 2: Find the x-intercept.

Set $y = 0$:

$$(1 + 3k)x + 0 + 3 + 7k = 0$$

Solving for $x$:

$$x = \frac{-3 - 7k}{1 + 3k}$$

Step 3: Find the y-intercept.

Set $x = 0$:

$$0 + (2 + 4k)y + 3 + 7k = 0$$

Solving for $y$:

$$y = \frac{-3 - 7k}{2 + 4k}$$

Step 4: Equate the intercepts.

Since the x- and y-intercepts are equal:

$$\frac{-3 + 7k}{1 + 3k} = \frac{-3 + 7k}{2 + 4k}$$

Solving for $k$:

$$2 + 4k = 1 + 3k$$ $$k = -1$$

Substitute $k = -1$ into the equation:

$$(1 - 3)x + (2 - 4)y + 3 - 7 = 0$$

Simplifying:

$$-2x - 2y - 4 = 0$$

Dividing by $-2$:

$$x + y + 2 = 0$$

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Key Formulas or Methods Used

• Point-slope form: $y - y_1 = m(x - x_1)$
• Slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$
• Equation of a line through two points.
• Conditions for parallel and perpendicular lines.