What are the points of intersection of the curve `4x^(2)-9y^(2)=1` with its conjugate axis?

Options

A. `((1)/(2), 0) and (-(1)/(2), 0) `
B. `(0, 2) and (0, -2)`
C. `(0, 3) and (0, -3)`
D. No such points exist

Answer

To find the points of intersection of the curve given by the equation \(4x^2 - 9y^2 = 1\) with its conjugate axis, we can follow these steps:

### Step 1: Identify the type of conic section
The equation \(4x^2 - 9y^2 = 1\) is a hyperbola. The standard form of a hyperbola is given by:
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
where \(a^2 = \frac{1}{4}\) and \(b^2 = \frac{1}{9}\).

### Step 2: Determine the values of \(a\) and \(b\)
From the given equation:
\[ a^2 = \frac{1}{4} \implies a = \frac{1}{2} \]
\[ b^2 = \frac{1}{9} \implies b = \frac{1}{3} \]

### Step 3: Identify the axes of the hyperbola
For the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\):
- The transverse axis is along the x-axis.
- The conjugate axis is along the y-axis.

### Step 4: Find the points of intersection with the conjugate axis
The conjugate axis corresponds to the line \(x = 0\). To find the points of intersection, we substitute \(x = 0\) into the hyperbola equation:
\[ 4(0)^2 - 9y^2 = 1 \implies -9y^2 = 1 \]
This simplifies to:
\[ y^2 = -\frac{1}{9} \]
Since \(y^2\) cannot be negative, there are no real solutions for \(y\).

### Conclusion
Thus, there are no points of intersection of the curve \(4x^2 - 9y^2 = 1\) with its conjugate axis.

### Final Answer
No such point exists.

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