Answer :
Sure! Let's solve this problem step by step.
### Part 1: [tex]\(L_1 \perp L_2\)[/tex]
1. Find the Slope of [tex]\(L_2\)[/tex]:
- [tex]\(L_2\)[/tex] makes an angle of 450 degrees with the positive x-axis. Since angles are typically measured between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex], we should convert 450 degrees by subtracting 360. This gives us [tex]\(450^\circ - 360^\circ = 90^\circ\)[/tex].
- An angle of 90 degrees indicates a vertical line, which means the slope of [tex]\(L_2\)[/tex] is undefined.
2. Perpendicular Condition:
- For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. However, since the slope of [tex]\(L_2\)[/tex] is undefined, the slope of [tex]\(L_1\)[/tex] must be 0, which means [tex]\(L_1\)[/tex] is a horizontal line.
3. Finding [tex]\(k\)[/tex] for Perpendicular Lines:
- The slope of a horizontal line [tex]\(L_1\)[/tex] is 0. This means the difference in y-coordinates must be zero: [tex]\((k - 1) = 0\)[/tex].
- Therefore, [tex]\(k = 1\)[/tex].
### Part 2: [tex]\(L_1 // L_2\)[/tex]
1. Find the Slope of [tex]\(L_2\)[/tex]:
- As established above, [tex]\(L_2\)[/tex] is a vertical line, with an undefined slope.
2. Parallel Condition:
- For lines to be parallel, they must have the same slope. But since a vertical line has an undefined slope, [tex]\(L_1\)[/tex] cannot have a defined slope if it were to be parallel.
- A horizontal line and a vertical line cannot be parallel. Therefore, there is no value of [tex]\(k\)[/tex] that would make [tex]\(L_1\)[/tex] parallel to [tex]\(L_2\)[/tex].
In summary:
- If [tex]\(L_1\)[/tex] is perpendicular to [tex]\(L_2\)[/tex], then [tex]\(k = 1\)[/tex].
- If [tex]\(L_1\)[/tex] is parallel to [tex]\(L_2\)[/tex], there is no solution (no such value of [tex]\(k\)[/tex]).
### Part 1: [tex]\(L_1 \perp L_2\)[/tex]
1. Find the Slope of [tex]\(L_2\)[/tex]:
- [tex]\(L_2\)[/tex] makes an angle of 450 degrees with the positive x-axis. Since angles are typically measured between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex], we should convert 450 degrees by subtracting 360. This gives us [tex]\(450^\circ - 360^\circ = 90^\circ\)[/tex].
- An angle of 90 degrees indicates a vertical line, which means the slope of [tex]\(L_2\)[/tex] is undefined.
2. Perpendicular Condition:
- For two lines to be perpendicular, the product of their slopes should be [tex]\(-1\)[/tex]. However, since the slope of [tex]\(L_2\)[/tex] is undefined, the slope of [tex]\(L_1\)[/tex] must be 0, which means [tex]\(L_1\)[/tex] is a horizontal line.
3. Finding [tex]\(k\)[/tex] for Perpendicular Lines:
- The slope of a horizontal line [tex]\(L_1\)[/tex] is 0. This means the difference in y-coordinates must be zero: [tex]\((k - 1) = 0\)[/tex].
- Therefore, [tex]\(k = 1\)[/tex].
### Part 2: [tex]\(L_1 // L_2\)[/tex]
1. Find the Slope of [tex]\(L_2\)[/tex]:
- As established above, [tex]\(L_2\)[/tex] is a vertical line, with an undefined slope.
2. Parallel Condition:
- For lines to be parallel, they must have the same slope. But since a vertical line has an undefined slope, [tex]\(L_1\)[/tex] cannot have a defined slope if it were to be parallel.
- A horizontal line and a vertical line cannot be parallel. Therefore, there is no value of [tex]\(k\)[/tex] that would make [tex]\(L_1\)[/tex] parallel to [tex]\(L_2\)[/tex].
In summary:
- If [tex]\(L_1\)[/tex] is perpendicular to [tex]\(L_2\)[/tex], then [tex]\(k = 1\)[/tex].
- If [tex]\(L_1\)[/tex] is parallel to [tex]\(L_2\)[/tex], there is no solution (no such value of [tex]\(k\)[/tex]).
