Consider the graphs of` y = Ax^2 and y^2 + 3 = x^2 + 4y`, where A is a positive constant and `x,y in R`.Number of points in which the two graphs intersect, is

Explanation

To find the number of points of intersection between the graphs of \( y = Ax^2 \) and \( y^2 + 3 = x^2 + 4y \), we can follow these steps:

### Step 1: Rewrite the second equation
The second equation is given as:
\[ y^2 + 3 = x^2 + 4y \]
Rearranging this gives:
\[ y^2 - 4y + 3 = x^2 \]

### Step 2: Factor the quadratic in \( y \)
The left-hand side can be factored:
\[ (y - 2)^2 - 1 = x^2 \]
This simplifies to:
\[ (y - 2)^2 = x^2 + 1 \]

### Step 3: Recognize the hyperbola
The equation \((y - 2)^2 - x^2 = 1\) is the standard form of a hyperbola centered at \((0, 2)\) with a vertical transverse axis.

### Step 4: Substitute the parabola equation
Now substitute \( y = Ax^2 \) into the hyperbola equation:
\[ (Ax^2 - 2)^2 - x^2 = 1 \]

### Step 5: Expand and simplify
Expanding the left-hand side:
\[ (A^2x^4 - 4Ax^2 + 4) - x^2 = 1 \]
This simplifies to:
\[ A^2x^4 - (4A + 1)x^2 + 3 = 0 \]

### Step 6: Let \( z = x^2 \)
Let \( z = x^2 \). Then, the equation becomes:
\[ A^2z^2 - (4A + 1)z + 3 = 0 \]

### Step 7: Use the discriminant to find the number of solutions
The number of points of intersection depends on the discriminant of this quadratic equation. The discriminant \( D \) is given by:
\[ D = (4A + 1)^2 - 4 \cdot A^2 \cdot 3 \]
Calculating this gives:
\[ D = (4A + 1)^2 - 12A^2 \]
Expanding:
\[ D = 16A^2 + 8A + 1 - 12A^2 = 4A^2 + 8A + 1 \]

### Step 8: Analyze the discriminant
The discriminant \( D = 4A^2 + 8A + 1 \) is a quadratic in \( A \) and since the coefficient of \( A^2 \) is positive, it opens upwards. The discriminant is always positive for all \( A > 0 \).

### Step 9: Conclusion on the number of intersections
Since the discriminant is positive, there are two distinct real solutions for \( z \), which means there are two values of \( x^2 \). Each value of \( x^2 \) corresponds to two values of \( y \) (since \( y = Ax^2 \) is a function of \( x^2 \)). Therefore, the total number of intersection points is:
\[ 2 \times 2 = 4 \]

Thus, the number of points in which the two graphs intersect is **4**.

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