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The problem involves two straight lines, \(L_1: hx + ky + 24 = 0\) and \(L_2: 3x - 4y - 15h = 0\), which are perpendicular to each other. Additionally, the \(y\)-intercept of \(L_1\) is given as \(-8\), and we need to find the \(x\)-intercept of \(L_2\).

### Step-by-step solution:

#### 1. \(L_1: hx + ky + 24 = 0\)
We are told that the \(y\)-intercept of \(L_1\) is \(-8\). To find the \(y\)-intercept, we set \(x = 0\) in the equation of the line:

\[
h(0) + ky + 24 = 0 \quad \Rightarrow \quad ky = -24 \quad \Rightarrow \quad y = \frac{-24}{k}.
\]

Since the \(y\)-intercept is given as \(-8\), we set \(y = -8\):

\[
-8 = \frac{-24}{k} \quad \Rightarrow \quad k = 3.
\]

#### 2. \(L_2: 3x - 4y - 15h = 0\)
The second equation is given as \(L_2: 3x - 4y - 15h = 0\). The lines \(L_1\) and \(L_2\) are perpendicular, so their slopes must satisfy the condition that the product of their slopes is \(-1\).

- The slope of \(L_1: hx + ky + 24 = 0\) is \( \frac{-h}{k} = \frac{-h}{3}\).
- The slope of \(L_2: 3x - 4y - 15h = 0\) is \( \frac{3}{4}\).

For the lines to be perpendicular:

\[
\left(\frac{-h}{3}\right) \times \left(\frac{3}{4}\right) = -1 \quad \Rightarrow \quad \frac{-h}{4} = -1 \quad \Rightarrow \quad h = 4.
\]

#### 3. Find the \(x\)-intercept of \(L_2\)
To find the \(x\)-intercept of \(L_2\), we set \(y = 0\) in the equation \(3x - 4y - 15h = 0\):

\[
3x - 15h = 0 \quad \Rightarrow \quad 3x = 15h \quad \Rightarrow \quad x = \frac{15h}{3} = 5h.
\]

Since \(h = 4\):

\[
x = 5 \times 4 = 20.
\]

Thus, the \(x\)-intercept of \(L_2\) is \(\boxed{20}\).

### Summary:
- The answer is **D) 20**.

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. How do you find the slope of a line from its general equation?
2. Why do perpendicular lines have slopes whose product is \(-1\)?
3. Can you explain the significance of \(x\)-intercepts and \(y\)-intercepts in a graph?
4. How would the solution change if the lines were parallel instead of perpendicular?
5. What happens to the \(x\)-intercept if we change the value of \(h\)?

**Tip**: When two lines are perpendicular, their slopes are negative reciprocals, which means multiplying their slopes will always give \(-1\).

The straight lines L1: hx + ky + 24 = 0 and L2: 3x - 4y - 15h = 0 are perpendicular to each other. If the y-intercept of L1 is -8, then the x-intercept of L2 is?

Solve a problem involving perpendicular lines and their intercepts. Learn how to find the x-intercept of L2 when given the y-intercept of L1 and other parameters. Suitable for high school students in Grades 10-12.

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