4.3 Q-30

Question Statement

Find the distance from the point $P(6, -1)$ to the line $6x - 4y + 9 = 0$.

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Background and Explanation

The shortest distance from a point to a line is a perpendicular distance. To calculate this, we use the distance formula for a point and a line:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

where:

• $a$, $b$, and $c$ are coefficients from the line equation $ax + by + c = 0$,
• $(x_1, y_1)$ is the given point,
• $d$ is the perpendicular distance.

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Solution

Step 1: Write down the given values

• Point: $P(6, -1)$
• Line: $6x - 4y + 9 = 0$

From the line equation:

$$a = 6, \quad b = -4, \quad c = 9$$

Step 2: Apply the distance formula

The distance formula is:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

Substitute the given values:

$$d = \frac{|6(6) + (-4)(-1) + 9|}{\sqrt{6^2 + (-4)^2}}$$

Step 3: Simplify the numerator

First, calculate $6(6)$:

$$6(6) = 36$$

Next, calculate $(-4)(-1)$:

$$(-4)(-1) = 4$$

Add the terms in the numerator:

$$36 + 4 + 9 = 49$$

So, the numerator is $|49| = 49$.

Step 4: Simplify the denominator

Compute $6^2$ and $(-4)^2$:

$$6^2 = 36, \quad (-4)^2 = 16$$

Add these values:

$$36 + 16 = 52$$

Take the square root:

$$\sqrt{52} = 2\sqrt{13}$$

Step 5: Calculate the distance

Combine the numerator and denominator:

$$d = \frac{49}{2\sqrt{13}}$$

This is the required distance.

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Key Formulas or Methods Used

• Distance formula (Point to Line):

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

• Simplification of algebraic expressions.

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Summary of Steps

1. Identify $a$, $b$, $c$ from the line equation and the coordinates $(x_1, y_1)$ of the point.
2. Substitute the values into the distance formula:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

3. Simplify the numerator and denominator step-by-step.
4. Finalize the result as $d = \frac{49}{2\sqrt{13}}$.