Find the equations of the circles which touch Y-axis at
the point (0,3) , and make an intercept of 8 units on the X-axis .

Answer

To find the equations of the circles that touch the Y-axis at the point (0, 3) and make an intercept of 8 units on the X-axis, we can follow these steps:

### Step 1: Understand the Circle's Properties
The circle touches the Y-axis at (0, 3). This means that the distance from the center of the circle to the Y-axis is equal to the radius of the circle.

### Step 2: Determine the Center of the Circle
Let the center of the circle be at point (h, k). Since the circle touches the Y-axis at (0, 3), the Y-coordinate of the center (k) must be 3. Thus, we have:
- k = 3

The distance from the center (h, 3) to the Y-axis is |h|, which must equal the radius (r). Therefore, we can write:
- r = |h|

### Step 3: Analyze the X-axis Intercept
The circle makes an intercept of 8 units on the X-axis. This means that the distance from one point of intersection to the other on the X-axis is 8 units. Therefore, if the circle intersects the X-axis at points (a, 0) and (b, 0), we have:
- |a - b| = 8

Since the center is at (h, 3), the X-axis intercepts can be found using the radius:
- a = h - r
- b = h + r

Thus, we can express the intercept condition as:
- |(h - r) - (h + r)| = 8
- | -2r | = 8
- 2r = 8 or 2r = -8 (we take the positive value since radius cannot be negative)
- r = 4

### Step 4: Relate Radius to Center
From our earlier relationship, we know that:
- r = |h| = 4

This gives us two possible values for h:
- h = 4 or h = -4

### Step 5: Determine the Circle Equations
Now we have two centers:
1. Center (4, 3) with radius 4
2. Center (-4, 3) with radius 4

Using the standard form of the circle equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

For the first circle (center (4, 3)):
\[ (x - 4)^2 + (y - 3)^2 = 4^2 \\ (x - 4)^2 + (y - 3)^2 = 16 \]

For the second circle (center (-4, 3)):
\[ (x + 4)^2 + (y - 3)^2 = 4^2 \\ (x + 4)^2 + (y - 3)^2 = 16 \]

### Final Equations of the Circles
1. \((x - 4)^2 + (y - 3)^2 = 16\)
2. \((x + 4)^2 + (y - 3)^2 = 16\)

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