If the straight line `2x-3y+17=0` is perpendicular to the line passing through the points `(7, 17)` and `(15, beta)`, then `beta` equals

Options

A. `-5`
B. `-(35)/(3)`
C. `(35)/(3)`
D. 5

Answer

To solve the problem step by step, we need to find the value of \( \beta \) such that the line \( 2x - 3y + 17 = 0 \) is perpendicular to the line passing through the points \( (7, 17) \) and \( (15, \beta) \).

### Step 1: Find the slope of the given line
First, we need to rewrite the equation of the line \( 2x - 3y + 17 = 0 \) in the slope-intercept form \( y = mx + c \).

\[ 2x - 3y + 17 = 0 \implies 3y = 2x + 17 \implies y = \frac{2}{3}x + \frac{17}{3} \]

From this, we can see that the slope \( m_1 \) of the line is \( \frac{2}{3} \).

### Step 2: Find the slope of the line through the points
Next, we need to find the slope \( m_2 \) of the line that passes through the points \( (7, 17) \) and \( (15, \beta) \).

The formula for the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the points \( (7, 17) \) and \( (15, \beta) \):

\[ m_2 = \frac{\beta - 17}{15 - 7} = \frac{\beta - 17}{8} \]

### Step 3: Use the property of perpendicular lines
Since the two lines are perpendicular, the product of their slopes must equal \(-1\):

\[ m_1 \cdot m_2 = -1 \]

Substituting the values of \( m_1 \) and \( m_2 \):

\[ \frac{2}{3} \cdot \frac{\beta - 17}{8} = -1 \]

### Step 4: Solve for \( \beta \)
Now, we can solve for \( \beta \):

\[ \frac{2(\beta - 17)}{24} = -1 \]

Multiplying both sides by 24:

\[ 2(\beta - 17) = -24 \]

Dividing both sides by 2:

\[ \beta - 17 = -12 \]

Adding 17 to both sides:

\[ \beta = -12 + 17 = 5 \]

### Final Answer
Thus, the value of \( \beta \) is \( 5 \).

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