Locus of a point that divides a chord having slope 4 hyperbola xy=1 in the ratio `1 : 2`, is :

Options

A. `x^(2)+y^(2)=10`
B. `xy=1`
C. `x^(2)=4y`
D. `16x^(2)+10xy+y^(2)=2`

Explanation

To find the locus of a point that divides a chord of the hyperbola \( xy = 1 \) with a slope of 4 in the ratio \( 1:2 \), we can follow these steps:

### Step 1: Equation of the Chord
The slope of the chord is given as 4. Therefore, the equation of the line can be expressed as:
\[ y = 4x + c \]
where \( c \) is a constant.

### Step 2: Intersection with the Hyperbola
To find the points of intersection of the line with the hyperbola \( xy = 1 \), we substitute \( y \) from the line equation into the hyperbola equation:
\[ x(4x + c) = 1 \]
This simplifies to:
\[ 4x^2 + cx - 1 = 0 \]
Let the roots of this quadratic equation be \( t_1 \) and \( t_2 \).

### Step 3: Coordinates of Intersection Points
The coordinates of the intersection points \( A \) and \( B \) can be expressed as:
\[ A(t_1, \frac{1}{t_1}) \quad \text{and} \quad B(t_2, \frac{1}{t_2}) \]

### Step 4: Coordinates of the Point Dividing the Chord
Let \( P(h, k) \) be the point that divides the chord \( AB \) in the ratio \( 1:2 \). Using the section formula, the coordinates of point \( P \) are given by:
\[ h = \frac{2t_1 + t_2}{3}, \quad k = \frac{2\left(\frac{1}{t_1}\right) + \left(\frac{1}{t_2}\right)}{3} \]

### Step 5: Expressing \( k \) in Terms of \( h \)
Substituting \( t_2 = 3h - 2t_1 \) into the expression for \( k \):
\[ k = \frac{2\left(\frac{1}{t_1}\right) + \left(\frac{1}{3h - 2t_1}\right)}{3} \]

### Step 6: Finding the Relationship Between \( h \) and \( k \)
Using the relationship \( t_1 t_2 = -\frac{1}{4k} \) from the quadratic equation, we can express \( t_1 \) and \( t_2 \) in terms of \( h \) and \( k \) and substitute back to find a relationship between \( h \) and \( k \).

### Step 7: Final Equation of the Locus
After simplifying the expressions and substituting back, we arrive at the locus equation:
\[ 16h^2 + 10hk + k^2 - 2 = 0 \]

### Conclusion
The locus of the point that divides the chord of the hyperbola \( xy = 1 \) with slope 4 in the ratio \( 1:2 \) is given by:
\[ 16x^2 + 10xy + y^2 - 2 = 0 \]