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 the two lines and then calculate the distance between them. Here’s a step-by-step solution:

### Step 1: Identify the equations of the lines
We know the gradients (slopes) of the two lines:
- Line 1 has a gradient (m1) of 2.
- Line 2 has a gradient (m2) of 6.

Both lines intersect at the point (40, 30).

### Step 2: Write the equation of Line 1
Using the point-slope form of the equation of a line, which is given by:
\[ y - y_1 = m(x - x_1) \]
For Line 1:
- \( m = 2 \)
- \( (x_1, y_1) = (40, 30) \)

Substituting these values into the equation:
\[ y - 30 = 2(x - 40) \]
Expanding this:
\[ y - 30 = 2x - 80 \]
\[ y = 2x - 50 \]

### Step 3: Find the y-intercept of Line 1
To find the y-intercept, set \( x = 0 \):
\[ y = 2(0) - 50 = -50 \]
So, the y-intercept of Line 1 is -50.

### Step 4: Write the equation of Line 2
Using the same point-slope form for Line 2:
- \( m = 6 \)
- \( (x_1, y_1) = (40, 30) \)

Substituting these values into the equation:
\[ y - 30 = 6(x - 40) \]
Expanding this:
\[ y - 30 = 6x - 240 \]
\[ y = 6x - 210 \]

### Step 5: Find the y-intercept of Line 2
To find the y-intercept, set \( x = 0 \):
\[ y = 6(0) - 210 = -210 \]
So, the y-intercept of Line 2 is -210.

### Step 6: Calculate the distance between the y-intercepts
The distance between the y-intercepts is given by the absolute difference:
\[ \text{Distance} = |y_2 - y_1| = |-210 - (-50)| \]
Calculating this:
\[ = |-210 + 50| = |-160| = 160 \]

### Final Answer
The distance between the y-intercepts of the two lines is **160 units**.

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