Question

PQR is a right angled isosceles triangle with right angle at $ P(2, 1) $. If the equation of the line $ QR $ is $$ 2x + y = 3, $$ then the equation representing the pair of lines $ PQ $ and $ PR $ is:

• A. \( 3x^2 - 3y^2 - 8xy - 10x - 15y - 20 = 0 \)
• B. \( 3x^2 - 3y^2 + 8xy + 20x + 10y + 25 = 0 \)
• C. \( 3x^2 - 3y^2 + 8xy - 20x - 10y + 25 = 0 \)
• D. \( 3x^2 - 3y^2 + 8xy + 10x + 15y + 20 = 0 \)

Hint:

Use perpendicularity and point conditions to find pair of lines equations.

Answer 1

The Correct Option is C

1. Given \( P(2,1) \), and \( QR: 2x + y - 3 = 0 \). 2. Since triangle \( PQR \) is right-angled at \( P \), the lines \( PQ \) and \( PR \) are perpendicular and pass through \( P \). 3. Equation of \( PQ \) and \( PR \) is pair of lines passing through \( P \) and perpendicular to \( QR \). 4. Using formula for pair of lines through a point perpendicular to a given line, derive the equation: \[ 3x^2 - 3y^2 + 8xy - 20x - 10y + 25 = 0 \]