The locus of point from which the two tangents drawn to a parabola be such that slope of one is thrice of the other is

Options

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

Answer

To find the locus of a point from which two tangents are drawn to a parabola such that the slope of one tangent is three times the slope of the other, we can follow these steps:

### Step 1: Define the Parabola
We start with the standard form of the parabola, which is given by the equation:
\[ y^2 = 4ax \]

### Step 2: Equation of the Tangent
The equation of the tangent to the parabola from a point \( P(h, k) \) can be written as:
\[ y = mx + \frac{a}{m} \]
where \( m \) is the slope of the tangent.

### Step 3: Condition for the Point to Lie on the Tangent
Since the point \( P(h, k) \) lies on the tangent, we substitute \( P \) into the tangent equation:
\[ k = mh + \frac{a}{m} \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ mh - k + \frac{a}{m} = 0 \]
Multiplying through by \( m \) to eliminate the fraction:
\[ m^2h - km + a = 0 \]

### Step 5: Roots of the Quadratic Equation
This is a quadratic equation in \( m \). The slopes of the tangents are \( m_1 \) and \( m_2 \), where we know \( m_2 = 3m_1 \). Let \( m_1 = m \) and \( m_2 = 3m \).

### Step 6: Sum and Product of Roots
Using the properties of quadratic equations:
- The sum of the roots \( m + 3m = 4m \) is equal to:
\[ -\frac{-k}{h} = \frac{k}{h} \]
Thus, we have:
\[ 4m = \frac{k}{h} \]
- The product of the roots \( m \cdot 3m = 3m^2 \) is equal to:
\[ \frac{a}{h} \]

### Step 7: Expressing \( m \)
From \( 4m = \frac{k}{h} \), we can express \( m \):
\[ m = \frac{k}{4h} \]

### Step 8: Substitute \( m \) into Product of Roots
Now substituting \( m \) into the product of roots:
\[ 3\left(\frac{k}{4h}\right)^2 = \frac{a}{h} \]
This simplifies to:
\[ \frac{3k^2}{16h^2} = \frac{a}{h} \]

### Step 9: Rearranging the Equation
Cross-multiplying gives:
\[ 3k^2 = 16ah \]
Thus, we can express \( k^2 \):
\[ k^2 = \frac{16ah}{3} \]

### Step 10: Locus Equation
Now, replacing \( h \) and \( k \) with \( x \) and \( y \) respectively, we have:
\[ y^2 = \frac{16a}{3}x \]

### Final Result
The locus of the point from which the two tangents are drawn is given by:
\[ y^2 = \frac{16a}{3}x \]