Question

A rod of length eight units moves such that its ends A and B always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod AB internally in the ratio 2:1, is \[ 9(x^2 + \alpha y^2 + \beta xy + \gamma x + 28 y) - 76 = 0, \] then \[ \alpha - \beta - \gamma \text{ is equal to:} \]

• A. 24
• B. 23
• C. 21
• D. 22

Hint:

When dealing with geometric problems involving points dividing lines in a given ratio, use the section formula to find the coordinates of the dividing point. Simplify the resulting equations to identify the constants involved.

Answer 1

The Correct Option is B

Let the coordinates of point \( A \) be \( A(x_1, y_1) \) and the coordinates of point \( B \) be \( B(x_2, y_2) \). - The equation of line \( x - y + 2 = 0 \) gives the relationship for point \( A \), and the equation \( y + 2 = 0 \) gives the relationship for point \( B \). - Point \( A \) lies on the line \( x - y + 2 = 0 \), so we have the equation for \( A \): \[ y_1 = x_1 + 2. \] - Point \( B \) lies on the line \( y + 2 = 0 \), so \( y_2 = -2 \). The length of the rod is given as 8 units, so we apply the distance formula between points \( A \) and \( B \) to find the relation between \( x_1 \) and \( x_2 \). The distance between \( A(x_1, x_1+2) \) and \( B(x_2, -2) \) is 8 units: \[ \sqrt{(x_2 - x_1)^2 + (x_2 + 4)^2} = 8. \] - Solving for \( x_2 \) and \( x_1 \), we find the coordinates for points \( A \) and \( B \). - The point \( P \) divides the rod \( AB \) in the ratio 2:1, so using the section formula, the coordinates of \( P(x, y) \) are: \[ x = \frac{2x_2 + x_1}{3}, \quad y = \frac{2y_2 + y_1}{3}. \] - Using these expressions for \( x \) and \( y \), we substitute into the given equation for the locus of point \( P \), and after simplification, we find the values of \( \alpha \), \( \beta \), and \( \gamma \). - Finally, we calculate: \[ \alpha - \beta - \gamma = 23. \] Conclusion: The correct answer is (2), as \( \alpha - \beta - \gamma = 23 \).