Certainly! Let's address each part of the question step-by-step:
### i) Above the [tex]\( x \)[/tex]-axis, lying on the [tex]\( y \)[/tex]-axis, and at a distance of 6 units
1. Understanding the constraints:
- The point is above the [tex]\( x \)[/tex]-axis. This means its [tex]\( y \)[/tex]-coordinate is positive.
- The point is lying on the [tex]\( y \)[/tex]-axis. This means its [tex]\( x \)[/tex]-coordinate is 0.
2. Distance to the origin:
- The point is 6 units away from the origin.
- Since the point is on the [tex]\( y \)[/tex]-axis and 6 units above the origin, its coordinates are [tex]\((0, 6)\)[/tex].
So, the coordinates of the point are [tex]\((0, 6)\)[/tex].
### ii) Lying on the [tex]\( x \)[/tex]-axis to the left of the origin and at a distance of 3 units
1. Understanding the constraints:
- The point lies on the [tex]\( x \)[/tex]-axis. This means its [tex]\( y \)[/tex]-coordinate is 0.
- The point is to the left of the origin. This means its [tex]\( x \)[/tex]-coordinate is negative.
2. Distance to the origin:
- The point is 3 units away from the origin.
- Since it is to the left of the origin and on the [tex]\( x \)[/tex]-axis, the coordinates of the point are [tex]\((-3, 0)\)[/tex].
So, the coordinates of the point are [tex]\((-3, 0)\)[/tex].
### Summary
- For part i), the coordinates of the point are [tex]\((0, 6)\)[/tex].
- For part ii), the coordinates of the point are [tex]\((-3, 0)\)[/tex].
Here are the coordinates for both points:
1. [tex]\((0, 6)\)[/tex]
2. [tex]\((-3, 0)\)[/tex]
These accurately describe the points as per the conditions given in the question.
