If curve `x^2=9a(9-y)` and `x^2=a(y+1)` intersect orthogonally then value of 'a' is

Options

A. 3
B. 4
C. 5
D. 7

Explanation

To solve the problem of finding the value of 'a' such that the curves \( x^2 = 9a(9 - y) \) and \( x^2 = a(y + 1) \) intersect orthogonally, we can follow these steps:

### Step 1: Find the intersection points of the curves
We start by equating the two expressions for \( x^2 \):
\[ 9a(9 - y) = a(y + 1) \]
Assuming \( a \neq 0 \), we can divide both sides by \( a \):
\[ 9(9 - y) = y + 1 \]
Expanding this gives:
\[ 81 - 9y = y + 1 \]
Rearranging the equation:
\[ 81 - 1 = y + 9y \]
\[ 80 = 10y \]
Thus, we find:
\[ y = 8 \]

### Step 2: Substitute \( y \) back to find \( x^2 \)
Now substituting \( y = 8 \) back into either of the original equations to find \( x^2 \):
\[ x^2 = 9a(9 - 8) = 9a(1) = 9a \]
Thus, we have:
\[ x^2 = 9a \]
This gives us:
\[ x = \pm 3\sqrt{a} \]

### Step 3: Find the slopes of the tangents at the intersection points
Next, we need to calculate the slopes of the tangents to both curves at the intersection points.

**For the first curve:**
Differentiating \( x^2 = 9a(9 - y) \):
\[ 2x = -9a \frac{dy}{dx} \]
Thus, we have:
\[ \frac{dy}{dx} = -\frac{2x}{9a} \]
At the intersection points, substituting \( x = 3\sqrt{a} \):
\[ \frac{dy}{dx} = -\frac{2(3\sqrt{a})}{9a} = -\frac{6\sqrt{a}}{9a} = -\frac{2}{3\sqrt{a}} \]
Let this slope be \( m_1 \):
\[ m_1 = -\frac{2}{3\sqrt{a}} \]

**For the second curve:**
Differentiating \( x^2 = a(y + 1) \):
\[ 2x = a \frac{dy}{dx} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{2x}{a} \]
At the intersection points, substituting \( x = 3\sqrt{a} \):
\[ \frac{dy}{dx} = \frac{2(3\sqrt{a})}{a} = \frac{6\sqrt{a}}{a} = \frac{6}{\sqrt{a}} \]
Let this slope be \( m_2 \):
\[ m_2 = \frac{6}{\sqrt{a}} \]

### Step 4: Set the product of slopes equal to -1
Since the curves intersect orthogonally, we have:
\[ m_1 \cdot m_2 = -1 \]
Substituting the values of \( m_1 \) and \( m_2 \):
\[ \left(-\frac{2}{3\sqrt{a}}\right) \cdot \left(\frac{6}{\sqrt{a}}\right) = -1 \]
This simplifies to:
\[ -\frac{12}{3a} = -1 \]
Thus, we have:
\[ \frac{12}{3a} = 1 \]
Multiplying both sides by \( 3a \):
\[ 12 = 3a \]
Dividing both sides by 3:
\[ a = 4 \]

### Final Answer
The value of \( a \) is \( 4 \).

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