Equation ofa tangent passing through (2, 8) to the hyperbola `5x^(2) - y^(2) = 5` is

Options

A. `3x - y + 2= 0`
B. `23x - 3y -22 = 0`
C. `3x - 23y + 178 = 0`
D. `3x + y + 14 = 0`

Answer

To find the equation of the tangent to the hyperbola \(5x^2 - y^2 = 5\) that passes through the point \((2, 8)\), we can follow these steps:

### Step 1: Rewrite the hyperbola in standard form
The equation of the hyperbola is given as:
\[ 5x^2 - y^2 = 5 \]
Dividing the entire equation by 5, we get:
\[ \frac{x^2}{1} - \frac{y^2}{5} = 1 \]
This shows that \(a^2 = 1\) and \(b^2 = 5\).

### Step 2: Write the equation of the tangent line
The equation of the tangent to the hyperbola at any point can be expressed as:
\[ y = mx \pm \sqrt{a^2 m^2 - b^2} \]
Substituting \(a^2\) and \(b^2\):
\[ y = mx \pm \sqrt{m^2 - 5} \]

### Step 3: Substitute the point (2, 8) into the tangent equation
Since the tangent passes through the point \((2, 8)\), we substitute \(x = 2\) and \(y = 8\):
\[ 8 = 2m \pm \sqrt{m^2 - 5} \]
Rearranging gives:
\[ 8 - 2m = \pm \sqrt{m^2 - 5} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides results in:
\[ (8 - 2m)^2 = m^2 - 5 \]
Expanding the left side:
\[ 64 - 32m + 4m^2 = m^2 - 5 \]
Rearranging gives:
\[ 4m^2 - m^2 - 32m + 64 + 5 = 0 \]
This simplifies to:
\[ 3m^2 - 32m + 69 = 0 \]

### Step 5: Solve the quadratic equation for m
Using the quadratic formula \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ m = \frac{32 \pm \sqrt{(-32)^2 - 4 \cdot 3 \cdot 69}}{2 \cdot 3} \]
Calculating the discriminant:
\[ 1024 - 828 = 196 \]
Thus:
\[ m = \frac{32 \pm 14}{6} \]
Calculating the two possible values for \(m\):
1. \(m = \frac{46}{6} = \frac{23}{3}\)
2. \(m = \frac{18}{6} = 3\)

### Step 6: Find the equations of the tangents
1. For \(m = 3\):
\[ y = 3x \pm \sqrt{3^2 - 5} = 3x \pm \sqrt{4} = 3x \pm 2 \]
This gives two equations:
\[ y = 3x + 2 \quad \text{and} \quad y = 3x - 2 \]

2. For \(m = \frac{23}{3}\):
\[ y = \frac{23}{3}x \pm \sqrt{\left(\frac{23}{3}\right)^2 - 5} \]
Calculating the square root:
\[ \left(\frac{23}{3}\right)^2 = \frac{529}{9}, \quad 5 = \frac{45}{9} \quad \Rightarrow \quad \sqrt{\frac{484}{9}} = \frac{22}{3} \]
Thus:
\[ y = \frac{23}{3}x \pm \frac{22}{3} \]
This gives two equations:
\[ y = \frac{23}{3}x + \frac{22}{3} \quad \text{and} \quad y = \frac{23}{3}x - \frac{22}{3} \]

### Final Answer
The equations of the tangents passing through the point \((2, 8)\) to the hyperbola \(5x^2 - y^2 = 5\) are:
1. \(y = 3x + 2\)
2. \(y = 3x - 2\)
3. \(y = \frac{23}{3}x + \frac{22}{3}\)
4. \(y = \frac{23}{3}x - \frac{22}{3}\)