If the line through the points (h, 7) and (2, 3) intersects the line `3x-4y-5=0` at right angles, then the value of h is

Options

A. `-1`
B. 1
C. 5
D. `-5`

Answer

To solve the problem step by step, we need to find the value of \( h \) such that the line through the points \( (h, 7) \) and \( (2, 3) \) intersects the line given by the equation \( 3x - 4y - 5 = 0 \) at right angles.

### Step 1: Find the slope of the line through the points \( (h, 7) \) and \( (2, 3) \).

The formula for the slope \( m_1 \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the points \( (h, 7) \) and \( (2, 3) \):

\[ m_1 = \frac{3 - 7}{2 - h} = \frac{-4}{2 - h} \]

### Step 2: Find the slope of the line given by the equation \( 3x - 4y - 5 = 0 \).

We need to convert this equation into the slope-intercept form \( y = mx + c \).

Starting with:

\[ 3x - 4y - 5 = 0 \]

Rearranging gives:

\[ 4y = 3x - 5 \]

Dividing by 4:

\[ y = \frac{3}{4}x - \frac{5}{4} \]

Thus, the slope \( m_2 \) of this line is:

\[ m_2 = \frac{3}{4} \]

### Step 3: Use the condition for perpendicular lines.

Two lines are perpendicular if the product of their slopes is \( -1 \). Thus, we have:

\[ m_1 \cdot m_2 = -1 \]

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

\[ \left(\frac{-4}{2 - h}\right) \cdot \left(\frac{3}{4}\right) = -1 \]

### Step 4: Simplify and solve for \( h \).

Multiplying both sides by \( 4(2 - h) \) to eliminate the fraction:

\[ -4 \cdot 3 = -1 \cdot 4(2 - h) \]

This simplifies to:

\[ -12 = -4(2 - h) \]

Distributing the \( -4 \):

\[ -12 = -8 + 4h \]

Adding 8 to both sides:

\[ -12 + 8 = 4h \]

\[ -4 = 4h \]

Dividing by 4:

\[ h = -1 \]

### Final Answer:

The value of \( h \) is \( -1 \).

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