To solve the problem, we need to analyze the statements regarding the line \( L \) and the plane given by the equation \( 3x - 4y + 5z = 10 \).
### Step 1: Identify the normal vector of the plane
The equation of the plane can be expressed in the form \( Ax + By + Cz = D \), where \( A = 3 \), \( B = -4 \), and \( C = 5 \). The normal vector \( \mathbf{n} \) to the plane is given by the coefficients of \( x \), \( y \), and \( z \):
\[
\mathbf{n} = \langle 3, -4, 5 \rangle
\]
**Hint:** The normal vector of a plane can be directly obtained from the coefficients of \( x \), \( y \), and \( z \) in its equation.
### Step 2: Determine the direction ratios of the line \( L \)
Since line \( L \) is perpendicular to the plane, its direction ratios will be the same as the components of the normal vector. Thus, the direction ratios of line \( L \) are:
\[
\langle 3, -4, 5 \rangle
\]
**Hint:** A line perpendicular to a plane has direction ratios equal to the normal vector of that plane.
### Step 3: Calculate the direction cosines of line \( L \)
The direction cosines \( l, m, n \) are calculated using the formula:
\[
l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}
\]
where \( a, b, c \) are the direction ratios. Here, \( a = 3 \), \( b = -4 \), and \( c = 5 \).
First, calculate \( a^2 + b^2 + c^2 \):
\[
a^2 + b^2 + c^2 = 3^2 + (-4)^2 + 5^2 = 9 + 16 + 25 = 50
\]
Now, calculate the square root:
\[
\sqrt{50} = 5\sqrt{2}
\]
Now, we can find the direction cosines:
\[
l = \frac{3}{5\sqrt{2}}, \quad m = \frac{-4}{5\sqrt{2}}, \quad n = \frac{5}{5\sqrt{2}} = \frac{1}{\sqrt{2}}
\]
**Hint:** To find direction cosines, divide each direction ratio by the magnitude of the vector formed by those ratios.
### Step 4: Compare with Statement-2
The direction cosines we found are:
\[
\left( \frac{3}{5\sqrt{2}}, \frac{-4}{5\sqrt{2}}, \frac{1}{\sqrt{2}} \right)
\]
This matches exactly with the direction cosines given in Statement-2.
### Conclusion
- **Statement-1** is true: A line \( L \) can indeed be perpendicular to the plane \( 3x - 4y + 5z = 10 \).
- **Statement-2** is also true: The direction cosines of line \( L \) are correctly stated.
Therefore, both statements are true, and Statement-2 provides a correct explanation for Statement-1.
**Final Answer:** Both Statement-1 and Statement-2 are true, and Statement-2 is a correct explanation of Statement-1.
