If two tangents drawn from the point P (h,k) to the parabola `y^2=8x` are such that the slope of one of the tangent is 3 times the slope of the other , then the locus of point P is

Options

A. `3y^2=16x`
B. `3y^2=8x`
C. `y^2=32x`
D. `3y^2=32x`

Answer

To find the locus of the point \( P(h, k) \) from which two tangents are drawn to the parabola \( y^2 = 8x \) such that the slope of one tangent is three times the slope of the other, we can follow these steps:

### Step 1: Identify the equation of the parabola
The given parabola is \( y^2 = 8x \). This can be rewritten in the standard form \( y^2 = 4ax \) where \( a = 2 \).

### Step 2: Write the equation of the tangent
The equation of the tangent to the parabola \( y^2 = 8x \) at a point with slope \( m \) is given by:
\[ y = mx + \frac{2}{m} \]
This can be rearranged to:
\[ my - mx - 2 = 0 \]
This is a linear equation in \( x \) and \( y \).

### Step 3: Substitute point \( P(h, k) \) into the tangent equation
Since point \( P(h, k) \) lies on the tangent, we substitute \( x = h \) and \( y = k \):
\[ mh - k - 2 = 0 \quad \text{(1)} \]

### Step 4: Use the condition on slopes
Let the slopes of the two tangents be \( m \) and \( 3m \). The tangents will satisfy the quadratic equation formed by substituting these slopes into the tangent equation.

### Step 5: Form the quadratic equation
The sum of the roots (slopes) of the quadratic equation is given by:
\[ m + 3m = 4m = -\frac{b}{a} = \frac{k}{h} \quad \text{(2)} \]

The product of the roots is given by:
\[ m \cdot 3m = 3m^2 = \frac{c}{a} = \frac{2}{h} \quad \text{(3)} \]

### Step 6: Solve for \( m \) using equations (2) and (3)
From equation (2):
\[ m = \frac{k}{4h} \]

Substituting \( m \) into equation (3):
\[ 3\left(\frac{k}{4h}\right)^2 = \frac{2}{h} \]
This simplifies to:
\[ 3\frac{k^2}{16h^2} = \frac{2}{h} \]

### Step 7: Cross-multiply and simplify
Cross-multiplying gives:
\[ 3k^2 = 32h \]

### Step 8: Replace \( h \) and \( k \) with \( x \) and \( y \)
Since \( h \) and \( k \) represent coordinates, we can replace them with \( x \) and \( y \):
\[ 3y^2 = 32x \]

### Final Result
The locus of the point \( P(h, k) \) is given by:
\[ 3y^2 = 32x \]