To find the equation of the hyperbola from the definition that hyperbola is the locus of a point which moves such that the difference of its distances from two fixed points is constant with the fixed point as foci

Answer

To find the equation of the hyperbola from the definition that hyperbola is the locus of a point which moves such that the difference of its distances from two fixed points (the foci) is constant, we can follow these steps:

### Step 1: Define the foci and the point
Let the foci of the hyperbola be \( F_1(c, 0) \) and \( F_2(-c, 0) \). Let \( P(x, y) \) be any point on the hyperbola.

### Step 2: Set up the distance equation
According to the definition of a hyperbola, the difference in distances from the point \( P \) to the foci is a constant, which we will denote as \( 2a \). Therefore, we can write:
\[ |PF_1 - PF_2| = 2a \]
This can be expressed as:
\[ PF_1 - PF_2 = 2a \]
or
\[ PF_2 - PF_1 = 2a \]

### Step 3: Apply the distance formula
Using the distance formula, we can express \( PF_1 \) and \( PF_2 \):
\[ PF_1 = \sqrt{(x - c)^2 + (y - 0)^2} = \sqrt{(x - c)^2 + y^2} \]
\[ PF_2 = \sqrt{(x + c)^2 + (y - 0)^2} = \sqrt{(x + c)^2 + y^2} \]
Thus, we have:
\[ \sqrt{(x - c)^2 + y^2} - \sqrt{(x + c)^2 + y^2} = 2a \]

### Step 4: Square both sides
To eliminate the square roots, we square both sides:
\[ \left( \sqrt{(x - c)^2 + y^2} - \sqrt{(x + c)^2 + y^2} \right)^2 = (2a)^2 \]
This expands to:
\[ (x - c)^2 + y^2 - 2\sqrt{((x - c)^2 + y^2)((x + c)^2 + y^2)} + (x + c)^2 + y^2 = 4a^2 \]

### Step 5: Simplify the equation
Combining like terms, we get:
\[ (x^2 - 2cx + c^2 + y^2) + (x^2 + 2cx + c^2 + y^2) - 4a^2 = 2\sqrt{((x - c)^2 + y^2)((x + c)^2 + y^2)} \]
This simplifies to:
\[ 2x^2 + 2c^2 + 2y^2 - 4a^2 = 2\sqrt{((x - c)^2 + y^2)((x + c)^2 + y^2)} \]

### Step 6: Isolate the square root
Dividing everything by 2 gives:
\[ x^2 + c^2 + y^2 - 2a^2 = \sqrt{((x - c)^2 + y^2)((x + c)^2 + y^2)} \]

### Step 7: Square again
Square both sides again to eliminate the square root:
\[ (x^2 + c^2 + y^2 - 2a^2)^2 = ((x - c)^2 + y^2)((x + c)^2 + y^2) \]

### Step 8: Expand and rearrange
After expanding both sides and rearranging, we will eventually arrive at the standard form of the hyperbola:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \( c^2 = a^2 + b^2 \).

### Final Equation
Thus, the general equation of the hyperbola is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]

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