If `4x^(2)+ py^(2) =45` and `x^(2)-4y^(2) =5` cut orthogonally, then the value of p is

Options

A. `1//9`
B. `1//3`
C. 3
D. 9

Explanation

To solve the problem, we need to find the value of \( p \) such that the curves \( 4x^2 + py^2 = 45 \) and \( x^2 - 4y^2 = 5 \) intersect orthogonally. This means that the product of their slopes at the points of intersection should equal \(-1\).

### Step-by-Step Solution:

1. **Differentiate the first curve**:
The first curve is given by:
\[ 4x^2 + py^2 = 45 \]
Differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(4x^2) + \frac{d}{dx}(py^2) = 0 \]
This gives:
\[ 8x + 2py \frac{dy}{dx} = 0 \]
Rearranging for \( \frac{dy}{dx} \):
\[ 2py \frac{dy}{dx} = -8x \implies \frac{dy}{dx} = -\frac{4x}{py} \quad \text{(1)} \]

2. **Differentiate the second curve**:
The second curve is given by:
\[ x^2 - 4y^2 = 5 \]
Differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(x^2) - \frac{d}{dx}(4y^2) = 0 \]
This gives:
\[ 2x - 8y \frac{dy}{dx} = 0 \]
Rearranging for \( \frac{dy}{dx} \):
\[ 8y \frac{dy}{dx} = 2x \implies \frac{dy}{dx} = \frac{x}{4y} \quad \text{(2)} \]

3. **Set the product of slopes to -1**:
For the curves to intersect orthogonally, we need:
\[ \left(-\frac{4x}{py}\right) \left(\frac{x}{4y}\right) = -1 \]
Simplifying this:
\[ -\frac{4x^2}{4py^2} = -1 \implies \frac{x^2}{py^2} = 1 \implies x^2 = py^2 \quad \text{(3)} \]

4. **Substitute into the first curve**:
Substitute \( x^2 = py^2 \) into the first curve:
\[ 4(py^2) + py^2 = 45 \]
This simplifies to:
\[ 4py^2 + py^2 = 45 \implies (4p + p)y^2 = 45 \implies 5py^2 = 45 \]
Thus:
\[ py^2 = 9 \quad \text{(4)} \]

5. **Substitute \( y^2 \) back into (3)**:
From equation (4), we have \( y^2 = \frac{9}{p} \). Substitute this into equation (3):
\[ x^2 = p\left(\frac{9}{p}\right) = 9 \]
Therefore:
\[ x^2 = 9 \implies x = \pm 3 \]

6. **Find \( y \)**:
Substitute \( x^2 = 9 \) into the second curve:
\[ 9 - 4y^2 = 5 \implies 4y^2 = 4 \implies y^2 = 1 \implies y = \pm 1 \]

7. **Find \( p \)**:
Now we have \( x^2 = 9 \) and \( y^2 = 1 \). Substitute these values into equation (3):
\[ 9 = p(1) \implies p = 9 \]

### Conclusion:
The value of \( p \) is \( 9 \).