The locus of `P(x,y)` such that `sqrt(x^2+y^2+8y+16)-sqrt(x^2+y^2-6x+9)=5,` is

Options

A. hyperbola
B. circle
C. finite line segment
D. infinite ray

Explanation

To solve the problem, we need to find the locus of the point \( P(x, y) \) such that

\[ \sqrt{x^2 + y^2 + 8y + 16} - \sqrt{x^2 + y^2 - 6x + 9} = 5. \]

### Step 1: Simplifying the Equation

Start by rewriting the terms under the square roots:

1. The first square root can be simplified as follows:
\[ \sqrt{x^2 + y^2 + 8y + 16} = \sqrt{x^2 + (y + 4)^2}. \]
This is because \( y^2 + 8y + 16 = (y + 4)^2 \).

2. The second square root can also be simplified:
\[ \sqrt{x^2 + y^2 - 6x + 9} = \sqrt{(x - 3)^2 + y^2}. \]
This is because \( x^2 - 6x + 9 = (x - 3)^2 \).

### Step 2: Rewrite the Equation

Now substitute these simplifications back into the original equation:

\[ \sqrt{x^2 + (y + 4)^2} - \sqrt{(x - 3)^2 + y^2} = 5. \]

### Step 3: Isolate One Square Root

Next, isolate one of the square root terms:

\[ \sqrt{x^2 + (y + 4)^2} = \sqrt{(x - 3)^2 + y^2} + 5. \]

### Step 4: Square Both Sides

Square both sides to eliminate the square roots:

\[ x^2 + (y + 4)^2 = \left(\sqrt{(x - 3)^2 + y^2} + 5\right)^2. \]

Expanding both sides:

1. Left side:
\[ x^2 + y^2 + 8y + 16. \]

2. Right side:
\[ (x - 3)^2 + y^2 + 10\sqrt{(x - 3)^2 + y^2} + 25. \]

### Step 5: Expand and Simplify

Now, expand the right side:

\[ (x - 3)^2 + y^2 + 10\sqrt{(x - 3)^2 + y^2} + 25 = x^2 - 6x + 9 + y^2 + 10\sqrt{(x - 3)^2 + y^2} + 25. \]

Combine like terms:

\[ x^2 + y^2 - 6x + 34 + 10\sqrt{(x - 3)^2 + y^2}. \]

### Step 6: Set the Equation

Now set the left side equal to the right side:

\[ x^2 + y^2 + 8y + 16 = x^2 + y^2 - 6x + 34 + 10\sqrt{(x - 3)^2 + y^2}. \]

### Step 7: Cancel and Rearrange

Cancel \( x^2 + y^2 \) from both sides:

\[ 8y + 16 = -6x + 34 + 10\sqrt{(x - 3)^2 + y^2}. \]

Rearranging gives:

\[ 10\sqrt{(x - 3)^2 + y^2} = 6x - 8y + 18. \]

### Step 8: Square Again

Square both sides again to eliminate the square root:

\[ 100((x - 3)^2 + y^2) = (6x - 8y + 18)^2. \]

### Step 9: Expand and Simplify

Expand both sides and simplify to get the equation of the locus.

### Step 10: Identify the Locus

After simplification, you should find that the locus is a line, which can be expressed in intercept form.

### Final Answer

The locus of \( P(x, y) \) is given by the equation of the line:

\[ \frac{x}{3} - \frac{y}{4} = 1. \]