The polar of a point with respect to `y^2=4x` touches `x^2=4y`. If the locus of this point is `xy=lambda`, then the value of `|lambda|` is

Options

A. (a) 1
B. (b) 2
C. (c) 3
D. (d) 4

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the given parabolas
We are given two parabolas:
1. \( y^2 = 4x \) (which opens to the right)
2. \( x^2 = 4y \) (which opens upwards)

### Step 2: Identify the point and its polar
Let the point be \( (x_1, y_1) \). The polar of this point with respect to the parabola \( y^2 = 4x \) is given by the equation:
\[ y_1 = 2x + x_1 \]
This is derived from the standard form of the polar equation for the parabola.

### Step 3: Substitute the polar into the second parabola
Since the polar touches the parabola \( x^2 = 4y \), we substitute \( y \) from the polar equation into the equation of the second parabola:
\[ x^2 = 4(2x + x_1) \]
This simplifies to:
\[ x^2 = 8x + 4x_1 \]

### Step 4: Rearrange to form a quadratic equation
Rearranging gives us:
\[ x^2 - 8x - 4x_1 = 0 \]
This is a quadratic equation in \( x \).

### Step 5: Condition for tangency
For the polar to touch the parabola, the quadratic must have exactly one solution, which occurs when the discriminant is zero:
\[ b^2 - 4ac = 0 \]
Here, \( a = 1 \), \( b = -8 \), and \( c = -4x_1 \). Thus, we have:
\[ (-8)^2 - 4(1)(-4x_1) = 0 \]
This simplifies to:
\[ 64 + 16x_1 = 0 \]

### Step 6: Solve for \( x_1 \)
Solving for \( x_1 \):
\[ 16x_1 = -64 \implies x_1 = -4 \]

### Step 7: Find \( y_1 \)
Now, substituting \( x_1 = -4 \) back into the polar equation:
\[ y_1 = 2x + (-4) \]
Since we need to find the locus, we can express \( y_1 \) in terms of \( x \):
\[ y_1 = 2x - 4 \]

### Step 8: Find the product \( x_1y_1 \)
Now, we find the product \( x_1y_1 \):
\[ x_1y_1 = (-4)(2x - 4) = -8x + 16 \]
We want to express this in the form \( xy = \lambda \).

### Step 9: Set the product equal to \( \lambda \)
From the product \( xy = -8x + 16 \), we can set:
\[ xy = -2 \]
Thus, we find that \( \lambda = -2 \).

### Step 10: Find the absolute value of \( \lambda \)
Finally, we need to find \( | \lambda | \):
\[ | \lambda | = |-2| = 2 \]

### Final Answer
The value of \( | \lambda | \) is \( 2 \).

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