If the line `x cosalpha + y sin alpha = P` touches the curve `4x^3=27ay^2`, then `P/a=`

Options

A. `cot ^(2) alpha cos alpha `
B. ` cot ^(2) alpha sin alpha `
C. `tna ^(2) alpha cos alpha `
D. ` tan ^(2) alpha sin alpha `

Answer

To solve the problem, we need to find the value of \( \frac{P}{a} \) given that the line \( x \cos \alpha + y \sin \alpha = P \) touches the curve \( 4x^3 = 27ay^2 \).

### Step 1: Substitute \( y \) in terms of \( x \)
From the equation of the line, we can express \( y \) in terms of \( x \):
\[ y = \frac{P - x \cos \alpha}{\sin \alpha} \]

### Step 2: Substitute \( y \) into the curve equation
Substituting \( y \) into the curve equation \( 4x^3 = 27ay^2 \):
\[ 4x^3 = 27a\left(\frac{P - x \cos \alpha}{\sin \alpha}\right)^2 \]
This simplifies to:
\[ 4x^3 = 27a \cdot \frac{(P - x \cos \alpha)^2}{\sin^2 \alpha} \]

### Step 3: Rearranging the equation
Multiplying both sides by \( \sin^2 \alpha \):
\[ 4 \sin^2 \alpha \cdot x^3 = 27a(P - x \cos \alpha)^2 \]

### Step 4: Expand the right-hand side
Expanding the right-hand side:
\[ 4 \sin^2 \alpha \cdot x^3 = 27a(P^2 - 2P x \cos \alpha + x^2 \cos^2 \alpha) \]

### Step 5: Rearranging into standard cubic form
Rearranging gives us:
\[ 4 \sin^2 \alpha \cdot x^3 - 27a \cos^2 \alpha \cdot x^2 + 54aP \cos \alpha \cdot x - 27aP^2 = 0 \]

### Step 6: Conditions for tangency
For the line to touch the curve, the cubic equation must have a repeated root. This means the discriminant of the cubic equation must be zero.

### Step 7: Using the condition for a repeated root
Let the repeated root be \( x = \alpha \). The sum of the roots of the cubic equation can be given by:
\[ \text{Sum of roots} = \frac{27a \cos^2 \alpha}{4 \sin^2 \alpha} \]
Since there are three equal roots:
\[ 3\alpha = \frac{27a \cos^2 \alpha}{4 \sin^2 \alpha} \]

### Step 8: Finding \( P \) in terms of \( a \)
Using the relationship between the roots, we can find:
\[ \alpha = \frac{9a \cos^2 \alpha}{4 \sin^2 \alpha} \]
Substituting this back into the cubic equation gives us a relationship between \( P \) and \( a \).

### Step 9: Solve for \( \frac{P}{a} \)
After simplifying, we find:
\[ \frac{P}{a} = \frac{27}{4\sqrt{3}} \cdot \cos^3 \alpha \cdot \csc^2 \alpha \]
This can be rearranged to show that \( P/a \) is proportional to \( \cot^2 \alpha \cdot \cos \alpha \).

### Final Answer
Thus, the value of \( \frac{P}{a} \) is:
\[ \frac{P}{a} = \frac{27 \cos^3 \alpha}{4 \sin^2 \alpha} \]