The two conics `y^(2)/b^(2)-x^(2)/a^(2)=1` and `y^(2)=-b/ax` intersect if and only if

Options

A. `0ltale1/2`
B. `0ltble1/2`
C. `b^(2)gta^(2)`
D. `b^(2)lta^(2)`

Answer

To find the condition under which the two conics intersect, we need to analyze the equations of the conics given:

1. The first conic is given by the equation:
\[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]
This is a hyperbola.

2. The second conic is given by the equation:
\[ y^2 = -\frac{b}{a} x \]
This is a parabola that opens to the left since the coefficient of \(x\) is negative.

### Step 1: Substitute the second conic into the first conic
To find the points of intersection, we can substitute \(y^2\) from the second conic into the first conic:
\[ \frac{-\frac{b}{a} x}{b^2} - \frac{x^2}{a^2} = 1 \]
This simplifies to:
\[ -\frac{1}{ab} x - \frac{x^2}{a^2} = 1 \]

### Step 2: Multiply through by \(-ab\) to eliminate the fraction
Multiplying the entire equation by \(-ab\) gives:
\[ x + \frac{ab}{a^2} x^2 = -ab \]
This can be rearranged to:
\[ \frac{b}{a} x^2 + x + ab = 0 \]

### Step 3: Identify the coefficients of the quadratic equation
The quadratic equation can be written in standard form as:
\[ \frac{b}{a} x^2 + x + ab = 0 \]
Here, the coefficients are:
- \(p = \frac{b}{a}\)
- \(q = 1\)
- \(r = ab\)

### Step 4: Calculate the discriminant
The discriminant \(D\) of a quadratic equation \(px^2 + qx + r = 0\) is given by:
\[ D = q^2 - 4pr \]
Substituting the coefficients:
\[ D = 1^2 - 4 \left(\frac{b}{a}\right)(ab) = 1 - 4b^2 \]

### Step 5: Set the discriminant greater than zero for intersection
For the two conics to intersect, the discriminant must be greater than zero:
\[ 1 - 4b^2 > 0 \]
This simplifies to:
\[ 1 > 4b^2 \]
or
\[ \frac{1}{4} > b^2 \]
Taking the square root gives:
\[ -\frac{1}{2} < b < \frac{1}{2} \]

### Conclusion
Thus, the two conics intersect if and only if:
\[ b \in \left(-\frac{1}{2}, \frac{1}{2}\right) \]