A line with gradient 2 intersects a line with gradient 6 at the point (40, 30). The distance between y - intercepts of these lines is

Options

A. 160
B. 180
C. 108
D. 120

Answer

To solve the problem, we need to find the y-intercepts of two lines with gradients (slopes) of 2 and 6 that intersect at the point (40, 30). Then, we will calculate the distance between these y-intercepts.

### Step-by-Step Solution:

**Step 1: Find the equation of the first line (slope = 2)**
The formula for the equation of a line in point-slope form is:
\[ y - y_1 = m(x - x_1) \]
where \( m \) is the slope and \( (x_1, y_1) \) is the point of intersection.

For the first line:
- Slope \( m_1 = 2 \)
- Point of intersection \( (x_1, y_1) = (40, 30) \)

Substituting these values into the equation:
\[ y - 30 = 2(x - 40) \]

**Step 2: Rearranging the equation**
Expanding the equation:
\[ y - 30 = 2x - 80 \]
\[ y = 2x - 50 \]

**Step 3: Find the y-intercept of the first line**
To find the y-intercept, set \( x = 0 \):
\[ y = 2(0) - 50 = -50 \]
Thus, the y-intercept \( y_1 = -50 \).

**Step 4: Find the equation of the second line (slope = 6)**
For the second line:
- Slope \( m_2 = 6 \)
- Point of intersection \( (x_1, y_1) = (40, 30) \)

Using the point-slope form again:
\[ y - 30 = 6(x - 40) \]

**Step 5: Rearranging the equation**
Expanding this equation:
\[ y - 30 = 6x - 240 \]
\[ y = 6x - 210 \]

**Step 6: Find the y-intercept of the second line**
To find the y-intercept, set \( x = 0 \):
\[ y = 6(0) - 210 = -210 \]
Thus, the y-intercept \( y_2 = -210 \).

**Step 7: Calculate the distance between the y-intercepts**
The distance between the y-intercepts \( y_1 \) and \( y_2 \) is given by:
\[ \Delta y = |y_2 - y_1| \]
Substituting the values:
\[ \Delta y = |-210 - (-50)| = |-210 + 50| = |-160| = 160 \]

### Final Answer:
The distance between the y-intercepts of the two lines is \( 160 \) units.

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