If the chord of contact of tangents from a point P to the parabola `y^2 = 4ax` touches the parabola `x^2= 4by` then the locus of P is

Options

A. circle
B. parabola
C. ellipse
D. hyperbola

Answer

To find the locus of the point \( P(x_1, y_1) \) from which tangents are drawn to the parabola \( y^2 = 4ax \) that also touches the parabola \( x^2 = 4by \), we can follow these steps:

### Step 1: Write the equation of the chord of contact

The chord of contact from the point \( P(x_1, y_1) \) to the parabola \( y^2 = 4ax \) is given by the equation:

\[ yy_1 = 2a(x + x_1) \]

### Step 2: Rearranging the equation

Rearranging the equation gives us:

\[ yy_1 - 2ax - 2ax_1 = 0 \]

### Step 3: Find the condition for this chord to touch the second parabola

For this chord to touch the parabola \( x^2 = 4by \), we need to express the tangent line in the form of the second parabola's equation. The equation of the tangent to the parabola \( x^2 = 4by \) at any point \( (x_0, y_0) \) is:

\[ xx_0 = 2b(y + y_0) \]

### Step 4: Set the two equations equal

Now, we need to find the conditions under which the chord of contact touches the second parabola. Thus, we equate the two equations:

\[ yy_1 - 2ax - 2ax_1 = 0 \quad \text{and} \quad xx_0 - 2b(y + y_0) = 0 \]

### Step 5: Solve for the slopes

From the chord of contact equation, we can express \( x \) in terms of \( y \):

\[ x = \frac{yy_1 - 2ax_1}{2a} \]

Substituting this into the tangent equation gives us a relationship between \( y_1 \), \( x_1 \), \( a \), and \( b \).

### Step 6: Find the relationship between \( y_1 \) and \( x_1 \)

After substituting and simplifying, we will find:

\[ \frac{y_1}{2a} = \frac{b}{x_1} \]

### Step 7: Rearranging to find the locus

Rearranging gives us:

\[ y_1 = \frac{2ab}{x_1} \]

### Step 8: Locus of point \( P \)

This equation represents a hyperbola in the \( x_1y_1 \) plane. Thus, the locus of point \( P \) is given by:

\[ xy = 2ab \]

### Final Answer

The locus of the point \( P \) is a hyperbola.

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