two tangents are drawn from the point (-2 -1) to the parabola y^2=4x. if alpha os the angle between them then tan alpha is
Answers
Given : two tangents are drawn from the point (-2 -1) to the parabola y^2=4x. alpha is the angle between them
To find : Tan alpha , Tan α
Solution :
y²=4x.
=> 2ydy/dx = 4
=> dy/dx = 2/y
Slope = 2/y
Let say Tangent Point are A & B
A = (Ax , Ay) & B = (Bx , By)
Slope = 2/Ay
Ax = (Ay)²/4
((Ay)²/4 , Ay)) & (-2 -1)
Slope = ( Ay + 1)/((Ay)²/4 + 2)
( Ay + 1)/((Ay)²/4 + 2) = 2/Ay
=> Ay² + Ay = Ay²/2 + 4
=> Ay²/2 + Ay - 4 = 0
=> Ay² + 2Ay - 8 = 0
=> Ay² + 4Ay - 2Ay - 8 = 0
=> Ay(Ay + 4) - 2(Ay + 4) = 0
=> (Ay - 2)(Ay + 4) = 0
=> Ay = 2 or Ay = - 4
Ax = 1 or Ax = 4
A = ( 1 , 2) & B = ( 4 , -4)
Slope of one Tangent = 1
Slope of another Tangent = -1/2
Angle between Tangents = α
=> Tan α = | (1 - (-1/2) )/ (1 + 1(-1/2)) |
=> Tan α = | (3/2) / (1/2 ) |
=> Tan α = 3
Tan α = 3
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Answer:
answer attached
Step-by-step explanation:
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