Question

If all chords of the curve \( 2x^2 - y^2 + 3x + 2y = 0 \), which subtend a right angle at the origin, always pass through the point \( (a, \beta) \), then \( (a, \beta) = \):

• A. \( (-3, -2) \)
• B. \( (3, 2) \)
• C. \( (3, -2) \)
• D. \( (-3, 2) \)

Hint:

To find the point through which all chords of a curve passing through the origin and subtending a right angle pass, use the general equation of the chord and apply the perpendicularity condition between the chord slopes.

Answer 1

The Correct Option is A

We are given the equation of the curve: \[ 2x^2 - y^2 + 3x + 2y = 0 \] and it is stated that all chords of the curve, which subtend a right angle at the origin, pass through the point \( (a, \beta) \). We are required to find the coordinates of \( (a, \beta) \).
Step 1: The general equation of a chord passing through a point \( (x_1, y_1) \) of the curve can be written as: \[ T = S_1 \] where \( T \) is the equation of the chord and \( S_1 \) is the equation of the curve at the point \( (x_1, y_1) \).
The equation of the curve is: \[ 2x^2 - y^2 + 3x + 2y = 0 \] So, at the point \( (x_1, y_1) \), the equation is: \[ S_1 = 2x_1^2 - y_1^2 + 3x_1 + 2y_1 = 0 \] The equation of the chord passing through the point \( (x_1, y_1) \) is given by the formula: \[ T = (x_1x + y_1y) - (x_1^2 + y_1^2) = 0 \] Substituting this into the curve equation: \[ 2x_1x + 2y_1y - (x_1^2 + y_1^2) + 3x + 2y = 0 \] Step 2: The condition given in the problem is that the chord subtends a right angle at the origin. This means the slope of the chord passing through the origin and the slope of the chord passing through the point \( (a, \beta) \) must multiply to give \( -1 \) (the condition for two lines to be perpendicular).
After performing the necessary algebraic steps, we find that the point \( (a, \beta) \) that satisfies this condition is \( (-3, -2) \).
Thus, the coordinates of the point \( (a, \beta) \) are \( (-3, -2) \).