If the ratio of transverse and conjugate axis of the hyperbola `x^2 / a^2 - y^2 / b^2 = 1` passing through the point `(1, 1)` is `1/3`, then its equation is

Options

A. `9x^(2) - y^(2) = 9 `
B. `x^(2) - 9y^(2) = 8 `
C. `9x^(2) - y^(2) = 8 `
D. `x^(2) - 9y^(2) = 9 `

Answer

To solve the problem, we need to find the equation of the hyperbola given the conditions. Let's break it down step by step.

### Step 1: Understand the given hyperbola equation
The standard form of the hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \(2a\) is the length of the transverse axis and \(2b\) is the length of the conjugate axis.

### Step 2: Use the ratio of the axes
We are given that the ratio of the transverse axis to the conjugate axis is \( \frac{1}{3} \). Therefore, we can write:
\[ \frac{2a}{2b} = \frac{1}{3} \]
This simplifies to:
\[ \frac{a}{b} = \frac{1}{3} \]
From this, we can express \(b\) in terms of \(a\):
\[ b = 3a \]

### Step 3: Substitute the point into the hyperbola equation
The hyperbola passes through the point \((1, 1)\). We substitute \(x = 1\) and \(y = 1\) into the hyperbola equation:
\[ \frac{1^2}{a^2} - \frac{1^2}{b^2} = 1 \]
This simplifies to:
\[ \frac{1}{a^2} - \frac{1}{b^2} = 1 \]

### Step 4: Substitute \(b\) in terms of \(a\)
Now we substitute \(b = 3a\) into the equation:
\[ \frac{1}{a^2} - \frac{1}{(3a)^2} = 1 \]
This becomes:
\[ \frac{1}{a^2} - \frac{1}{9a^2} = 1 \]
Finding a common denominator, we have:
\[ \frac{9 - 1}{9a^2} = 1 \]
which simplifies to:
\[ \frac{8}{9a^2} = 1 \]

### Step 5: Solve for \(a^2\)
Cross-multiplying gives:
\[ 8 = 9a^2 \]
Thus, we find:
\[ a^2 = \frac{8}{9} \]

### Step 6: Find \(b^2\)
Now we can find \(b^2\) using \(b = 3a\):
\[ b^2 = (3a)^2 = 9a^2 = 9 \cdot \frac{8}{9} = 8 \]

### Step 7: Write the equation of the hyperbola
Now we substitute \(a^2\) and \(b^2\) back into the hyperbola equation:
\[ \frac{x^2}{\frac{8}{9}} - \frac{y^2}{8} = 1 \]
Multiplying through by \(72\) (the least common multiple of the denominators) to eliminate the fractions:
\[ 9x^2 - 9y^2 = 72 \]
Rearranging gives:
\[ 9x^2 - y^2 = 72 \]

### Final Step: Simplify the equation
The final equation of the hyperbola is:
\[ 9x^2 - y^2 = 72 \]