Find the co-ordinates of the points on the y-axis, which are at a distance of 10 units from the point (-8, 4)

Answer

To find the coordinates of the points on the y-axis that are at a distance of 10 units from the point (-8, 4), we can follow these steps:

### Step 1: Understand the coordinates on the y-axis
Since we are looking for points on the y-axis, the x-coordinate of any point on the y-axis is 0. Therefore, we can denote the point on the y-axis as (0, y).

### Step 2: Use the distance formula
The distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In our case, we have point A as (-8, 4) and point B as (0, y). The distance \( d \) is given as 10 units.

### Step 3: Substitute the coordinates into the distance formula
Substituting the coordinates into the distance formula, we have:
\[ 10 = \sqrt{(0 - (-8))^2 + (y - 4)^2} \]
This simplifies to:
\[ 10 = \sqrt{(8)^2 + (y - 4)^2} \]
\[ 10 = \sqrt{64 + (y - 4)^2} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 10^2 = 64 + (y - 4)^2 \]
\[ 100 = 64 + (y - 4)^2 \]

### Step 5: Isolate the squared term
Subtract 64 from both sides:
\[ 100 - 64 = (y - 4)^2 \]
\[ 36 = (y - 4)^2 \]

### Step 6: Take the square root of both sides
Taking the square root of both sides gives:
\[ y - 4 = \pm 6 \]

### Step 7: Solve for y
Now we have two equations to solve:
1. \( y - 4 = 6 \)
2. \( y - 4 = -6 \)

For the first equation:
\[ y = 6 + 4 = 10 \]

For the second equation:
\[ y = -6 + 4 = -2 \]

### Step 8: Write the final coordinates
The coordinates of the points on the y-axis that are at a distance of 10 units from the point (-8, 4) are:
1. \( (0, 10) \)
2. \( (0, -2) \)

### Final Answer
The coordinates of the points are \( (0, 10) \) and \( (0, -2) \).

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