If a line y = 3x + 1 cuts the parabola x2 - 4x - 4y + 20 = 0 at A and B, then the tangent of the angle subtended by line segment AB at the origin is

Question

If a line y = 3x + 1 cuts the parabola x2 - 4x - 4y + 20 = 0 at A and B, then the tangent of the angle subtended by line segment AB at the origin is (A) 8√3/205 (B) 8√3/209 (C) 8√3/215 (D) none of these

Tardigrade Admin · Accepted Answer

The joint equation of OA and OB is x2 - 4x (y - 3x) - 4y ( y - 3x) + 20(y - 3x)2 = 0 Making the equation of the parabola homogeneous using straight line, we get x2(1 + 12 + 180) - y2( 4 - 20) - xy ( 4 - 12 + 120) = 0 or 193x2 + 16y2 - 112 xy = 0 tan θ = (2√h2-ab/a+b) = (2√562 - 193 × 16/193 + 16) =(8√3/209)

Q. If a line $y = 3 x + 1$ cuts the parabola $x^{2} - 4 x - 4 y + 20 = 0$ at $A$ and $B$ , then the tangent of the angle subtended by line segment $A B$ at the origin is

$83 /205$

$83 /209$

$83 /215$

none of these

Q. If a line y=3x+1 cuts the parabola x2−4x−4y+20=0 at A and B, then the tangent of the angle subtended by line segment AB at the origin is

83​/205

83​/209

83​/215

none of these

The joint equation of OA and OB is x2−4x(y−3x)−4y(y−3x)+20(y−3x)2=0 Making the equation of the parabola homogeneous using straight line, we get x2(1+12+180)−y2(4−20)−xy(4−12+120)=0 or 193x2+16y2−112xy=0 tanθ=a+b2h2−ab​​ =193+162562−193×16​​=20983​​

Q. If a line $y = 3 x + 1$ cuts the parabola $x^{2} - 4 x - 4 y + 20 = 0$ at $A$ and $B$ , then the tangent of the angle subtended by line segment $A B$ at the origin is

$83 /205$

$83 /209$

$83 /215$