Let `O(0,0)` and `A(0, 1)` be two fixed points.Then the locus of a point P such that the perimeter of `DeltaAOP` is 4, is

Options

A. `9x^2+8y^2-8y=16`
B. `9x^2+8y^2+8y=16`
C. `8x^2+9y^2-9y=18`
D. `8x^2+9y^2+9y=18`

Explanation

To find the locus of the point \( P \) such that the perimeter of triangle \( AOP \) is 4, we will follow these steps:

### Step 1: Define the points
Let the coordinates of point \( O \) be \( (0, 0) \) and point \( A \) be \( (0, 1) \). Let the coordinates of point \( P \) be \( (h, k) \).

### Step 2: Write the perimeter condition
The perimeter of triangle \( AOP \) is given by:
\[ AO + OP + PA = 4 \]
Where:
- \( AO \) is the distance from \( A \) to \( O \),
- \( OP \) is the distance from \( O \) to \( P \),
- \( PA \) is the distance from \( P \) to \( A \).

### Step 3: Calculate the distances
Using the distance formula, we can express these distances:
1. \( AO = \sqrt{(0 - 0)^2 + (1 - 0)^2} = 1 \)
2. \( OP = \sqrt{(h - 0)^2 + (k - 0)^2} = \sqrt{h^2 + k^2} \)
3. \( PA = \sqrt{(h - 0)^2 + (k - 1)^2} = \sqrt{h^2 + (k - 1)^2} \)

### Step 4: Substitute distances into the perimeter equation
Substituting these distances into the perimeter equation:
\[ 1 + \sqrt{h^2 + k^2} + \sqrt{h^2 + (k - 1)^2} = 4 \]
This simplifies to:
\[ \sqrt{h^2 + k^2} + \sqrt{h^2 + (k - 1)^2} = 3 \]

### Step 5: Isolate one of the square roots
Let’s isolate one of the square roots:
\[ \sqrt{h^2 + (k - 1)^2} = 3 - \sqrt{h^2 + k^2} \]

### Step 6: Square both sides
Squaring both sides gives:
\[ h^2 + (k - 1)^2 = (3 - \sqrt{h^2 + k^2})^2 \]
Expanding both sides:
\[ h^2 + (k^2 - 2k + 1) = 9 - 6\sqrt{h^2 + k^2} + (h^2 + k^2) \]

### Step 7: Simplify the equation
Cancel \( h^2 + k^2 \) from both sides:
\[ -2k + 1 = 9 - 6\sqrt{h^2 + k^2} \]
Rearranging gives:
\[ 6\sqrt{h^2 + k^2} = 8 - 2k \]
Dividing by 2:
\[ 3\sqrt{h^2 + k^2} = 4 - k \]

### Step 8: Square again
Squaring both sides again:
\[ 9(h^2 + k^2) = (4 - k)^2 \]
Expanding the right side:
\[ 9h^2 + 9k^2 = 16 - 8k + k^2 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 9h^2 + 8k^2 + 8k - 16 = 0 \]

### Step 10: Replace variables
Replace \( h \) with \( x \) and \( k \) with \( y \):
\[ 9x^2 + 8y^2 - 8y - 16 = 0 \]

### Final Answer
The locus of the point \( P \) is given by the equation:
\[ 9x^2 + 8y^2 - 8y - 16 = 0 \]