Consider the circle `x^(2)+y^(2)=1` and thhe parabola `y=ax^(2)-b(agt0)`. This circle and parabola intersect at

Options

A. A.Four distinct points is `agtbgt1`
B. B.No point if `blt-1`
C. C.Two distinct points if `-1ltblt1`
D. D.One point if `b=1`

Explanation

To solve the problem of finding the intersection points of the circle \( x^2 + y^2 = 1 \) and the parabola \( y = ax^2 - b \) (where \( a > 0 \)), we will follow these steps:

### Step 1: Substitute the equation of the parabola into the circle's equation.
We start with the equations:
- Circle: \( x^2 + y^2 = 1 \)
- Parabola: \( y = ax^2 - b \)

Substituting the expression for \( y \) from the parabola into the circle's equation gives:
\[ x^2 + (ax^2 - b)^2 = 1 \]

### Step 2: Expand the equation.
Now, we expand the equation:
\[ x^2 + (ax^2 - b)^2 = x^2 + (a^2x^4 - 2abx^2 + b^2) = 1 \]
This simplifies to:
\[ x^2 + a^2x^4 - 2abx^2 + b^2 = 1 \]

### Step 3: Rearrange the equation.
Rearranging gives:
\[ a^2x^4 + (1 - 2ab)x^2 + (b^2 - 1) = 0 \]
Let \( t = x^2 \). The equation becomes:
\[ a^2t^2 + (1 - 2ab)t + (b^2 - 1) = 0 \]

### Step 4: Find the discriminant.
To determine the nature of the roots, we calculate the discriminant \( D \):
\[ D = (1 - 2ab)^2 - 4a^2(b^2 - 1) \]

### Step 5: Analyze the discriminant.
We need to analyze the discriminant \( D \):
1. If \( D > 0 \): There are two distinct real roots for \( t \), which means there are four distinct real values for \( x \) (since \( x = \pm \sqrt{t} \)).
2. If \( D = 0 \): There is one real root for \( t \), which means there are two distinct real values for \( x \).
3. If \( D < 0 \): There are no real roots for \( t \), which means there are no real values for \( x \).

### Step 6: Conclusion based on conditions.
- If \( b \leq -1 \): The discriminant \( D \) is less than 0, leading to no intersection points.
- If \( 1 < b < -1 \): The discriminant \( D \) is equal to 0, leading to two intersection points.
- If \( b > 1 \): The discriminant \( D \) is greater than 0, leading to four intersection points.

### Summary of Results:
- **No intersection points** if \( b \leq -1 \).
- **Two intersection points** if \( 1 < b < -1 \).
- **Four intersection points** if \( b > 1 \).