The curve passing through `P(pi^(2),pi)` is such that for a tangent drawn to it at a point Q, the ratio of the y - intercept and the ordinate of Q is `1:2`. Then, the equation of the curve is

Options

A. `y=pix^(2)`
B. `y=pisqrtx`
C. `y=sqrtx`
D. `y=pi^(2)x`

Answer

To find the equation of the curve that passes through the point \( P(\pi^2, \pi) \) and satisfies the given condition regarding the tangent at point \( Q \), we can follow these steps:

### Step 1: Understand the Given Condition
We know that for a tangent drawn to the curve at point \( Q(p, q) \), the ratio of the y-intercept of the tangent to the ordinate (y-coordinate) of point \( Q \) is \( 1:2 \).

### Step 2: Write the Equation of the Tangent
The equation of the tangent to the curve at point \( Q(p, q) \) can be expressed as:
\[ y - q = m(x - p) \]
where \( m \) is the slope of the tangent.

### Step 3: Find the Y-Intercept of the Tangent
To find the y-intercept, we set \( x = 0 \):
\[ y - q = m(0 - p) \implies y = -mp + q \]
Thus, the y-intercept is \( -mp + q \).

### Step 4: Set Up the Ratio Condition
According to the problem, the ratio of the y-intercept to the y-coordinate \( q \) is \( 1:2 \):
\[ \frac{-mp + q}{q} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2(-mp + q) = q \implies -2mp + 2q = q \]
Rearranging this, we have:
\[ -2mp + q = 0 \implies q = 2mp \]

### Step 5: Express the Slope in Terms of Derivative
The slope \( m \) of the tangent at point \( Q \) can be expressed as:
\[ m = \frac{dq}{dp} \]
Substituting this into our equation gives:
\[ q = 2p \frac{dq}{dp} \]

### Step 6: Rearranging and Integrating
Rearranging the equation:
\[ \frac{dq}{q} = \frac{2}{p} dp \]
Now, we integrate both sides:
\[ \int \frac{1}{q} dq = \int \frac{2}{p} dp \]
This results in:
\[ \ln |q| = 2 \ln |p| + C \]
Exponentiating both sides gives:
\[ q = k p^2 \quad \text{(where \( k = e^C \))} \]

### Step 7: Use the Point \( P(\pi^2, \pi) \) to Find \( k \)
Since the curve passes through \( P(\pi^2, \pi) \):
\[ \pi = k (\pi^2)^2 = k \pi^4 \]
Thus:
\[ k = \frac{\pi}{\pi^4} = \frac{1}{\pi^3} \]

### Step 8: Write the Final Equation of the Curve
Substituting \( k \) back into the equation:
\[ q = \frac{1}{\pi^3} p^2 \]
Replacing \( p \) with \( x \) and \( q \) with \( y \):
\[ y = \frac{1}{\pi^3} x^2 \]

### Conclusion
The equation of the curve is:
\[ y = \frac{1}{\pi^3} x^2 \]