Focus of hyperbola is `(+-3,0)` and equation of tangent is `2x+y-4=0`, find the equation of hyperbola is

Options

A. `4x^(2)-5y^(2)=20`
B. `5x^(2)-4y^(2)=20`
C. `4x^(2)-5y^(2)=1`
D. `5x^(2)-4y^(2)=1`

Answer

To find the equation of the hyperbola given its foci and the equation of a tangent, we can follow these steps:

### Step 1: Identify the foci and parameters
The foci of the hyperbola are given as \( (±3, 0) \). This indicates that the hyperbola is centered at the origin and opens along the x-axis. The distance from the center to each focus is denoted as \( c \), so we have:
\[ c = 3 \]

### Step 2: Relate \( c \), \( a \), and \( b \)
For hyperbolas, the relationship between \( a \), \( b \), and \( c \) is given by:
\[ c^2 = a^2 + b^2 \]
Since \( c = 3 \), we can write:
\[ c^2 = 9 \]
Thus, we have:
\[ 9 = a^2 + b^2 \quad \text{(Equation 1)} \]

### Step 3: Use the tangent equation
The equation of the tangent is given as \( 2x + y - 4 = 0 \). We can rewrite this in slope-intercept form:
\[ y = -2x + 4 \]
From this, we can identify the slope \( m = -2 \) and the y-intercept \( c = 4 \).

### Step 4: Use the tangent condition
For hyperbolas, the relationship involving the slope of the tangent line is given by:
\[ c^2 = a^2 m^2 - b^2 \]
Substituting the values we have:
\[ 16 = a^2 (-2)^2 - b^2 \]
This simplifies to:
\[ 16 = 4a^2 - b^2 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations
Now we have two equations:
1. \( 9 = a^2 + b^2 \) (Equation 1)
2. \( 16 = 4a^2 - b^2 \) (Equation 2)

From Equation 1, we can express \( b^2 \) in terms of \( a^2 \):
\[ b^2 = 9 - a^2 \]
Substituting this into Equation 2:
\[ 16 = 4a^2 - (9 - a^2) \]
This simplifies to:
\[ 16 = 4a^2 - 9 + a^2 \]
\[ 16 + 9 = 5a^2 \]
\[ 25 = 5a^2 \]
\[ a^2 = 5 \]

### Step 6: Find \( b^2 \)
Now substituting \( a^2 = 5 \) back into Equation 1:
\[ 9 = 5 + b^2 \]
\[ b^2 = 4 \]

### Step 7: Write the equation of the hyperbola
The standard form of the hyperbola that opens along the x-axis is:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Substituting \( a^2 = 5 \) and \( b^2 = 4 \):
\[ \frac{x^2}{5} - \frac{y^2}{4} = 1 \]
To eliminate the fractions, we can multiply through by 20:
\[ 4x^2 - 5y^2 = 20 \]

### Final Answer
The equation of the hyperbola is:
\[ 4x^2 - 5y^2 = 20 \]