The problem you've provided appears to involve two lines and the geometric relationships between them. Here's the breakdown of the question:

1. Given Information:

   • Line $L_1$ has the equation $5x + y - 2 = 0$.
   • Line $L_2$ cuts the y-axis at $M(0, 1)$ and is parallel to $L_1$.
   • Point $P$ lies on $L_2$ such that $MP = NP$, where $N$ has coordinates $(6, 5)$.

2. Part (a):

   • You are asked to find the equation of line $L_2$.

3. Part (b):

   • You are asked to find the coordinates of point $P$.

4. Part (c):

   • You are asked whether $\triangle MNP$ is a right-angled triangle and to explain your answer.

Step-by-Step Solution:

(a) Find the equation of $L_2$

Since $L_2$ is parallel to $L_1$, it will have the same slope. First, we determine the slope of $L_1$. The equation of $L_1$ is in the form: $5x + y - 2 = 0 \quad \Rightarrow \quad y = -5x + 2.$ So, the slope of $L_1$ is $-5$.

Since $L_2$ is parallel to $L_1$, it will also have a slope of $-5$. The y-intercept of $L_2$ is given as $M(0, 1)$, so the equation of $L_2$ is: $y = -5x + 1.$

(b) Find the coordinates of $P$

We are told that $MP = NP$, which means point $P$ is equidistant from $M(0, 1)$ and $N(6, 5)$.

To find $P$, we use the midpoint formula. The midpoint of $M(0, 1)$ and $N(6, 5)$ will lie on $L_2$. The midpoint $P$ is calculated as: $P = \left( \frac{0 + 6}{2}, \frac{1 + 5}{2} \right) = (3, 3).$ So, the coordinates of $P$ are $(3, 3)$.

(c) Is $\triangle MNP$ a right-angled triangle?

To check if $\triangle MNP$ is a right-angled triangle, we need to see if the angle between any two sides is $90^\circ$. This can be done by checking if the slopes of two sides are perpendicular (i.e., the product of the slopes is $-1$).

• Slope of $MN$ (between $M(0, 1)$ and $N(6, 5)$): $\text{slope of } MN = \frac{5 - 1}{6 - 0} = \frac{4}{6} = \frac{2}{3}.$

• Slope of $MP$ (between $M(0, 1)$ and $P(3, 3)$): $\text{slope of } MP = \frac{3 - 1}{3 - 0} = \frac{2}{3}.$

Since the slopes of $MN$ and $MP$ are the same, the angle between these two sides is not $90^\circ$. Therefore, $\triangle MNP$ is not a right-angled triangle.

Would you like further clarification or more details on any part of the solution?

Here are some related questions to explore further:

1. How do you find the distance between two points?
2. What is the significance of parallel lines in geometry?
3. Can the slopes of two lines help in identifying different types of triangles?
4. How can you determine if a triangle is isosceles based on point coordinates?
5. What are some other methods to confirm that a triangle is right-angled?

Tip: The product of slopes for two perpendicular lines is always $-1$.