AP EAMCET20224 Jul 2022Evening ShiftMathematicsStraight LinesActual
A line passing through P 2 , 3 and making an angle of 30 ° with the positive direction of x -axis meets x 2 - 2 x y - y 2 = 0 at A and B . Then the value of P A . P B is
Options
- A17 3 + 1
- B17 3 + 1
- C17 3 - 1
- D17 3 - 1
Correct answer
B. 17 3 + 1
Step-by-step solution
Given, Equation of the line through P 2 , 3 is So parametric equation of line is given by, x - 2 cos θ = y - 3 sin θ = r     where   θ = 30 ° Now if P A = r 1 ,   P B = r 2 then r 1 , r 2 are the roots of the equation x 2 - 2 x y - y 2 = 0 ⇒ 2 + r cos θ 2 - 2 2 + r cos θ 3 + r sin θ - 3 + r sin θ 2 = 0 ⇒ r 2 cos 2 θ - sin 2 θ - 2 r cos θ + 5 sin θ - 17 = 0 So P A · P B = r 1 r 2 = - 17 cos 2 θ - sin 2 θ = 17 3