acute angle between the curves x²+y²=50 and x²=5y is
Answers
EXPLANATION.
Acute angle between the curves,
⇒ x² + y² = 50 ⇒ (1).
⇒ x² = 5y ⇒ (2).
Put the value of equation (1) in equation (2), we get.
⇒ y² + 5y = 50.
⇒ y² + 5y - 50 = 0.
⇒ y² + 10y - 5y - 50 = 0.
⇒ y (y + 10) - 5(y + 10).
⇒ (y - 5)(y + 10) = 0.
⇒ y = 5 and y = - 10.
Put the value of y in equation, we get.
Put y = 5 in equation, (2) we get.
⇒ x² = 5(5).
⇒ x² = 25.
⇒ x = √25.
⇒ x = ± 5.
⇒ x² + y² = 50.
Differentiate w.r.t x we get,
⇒ 2x + 2y.dy/dx = 0.
⇒ 2y.dy/dx = -2x.
⇒ dy/dx = -2x/2y.
⇒ dy/dx = -x/y.
Put x = -5 in equation, we get.
⇒ dy/dx = -(-5)/5.
⇒ dy/dx = 1. = m₁.
Put x = 5 in equation, we get.
⇒ dy/dx = -5/5.
⇒ dy/dx = -1 = m₁
M₁ = (1,-1).
⇒ x² = 5y.
Differentiate w.r.t x we get.
⇒ 2x = 5.dy/dx + y.
⇒ 2x = 5.dy/dx(y).
⇒ 2x/5y = dy/dx.
Put the value of x = -5 in equation, we get.
⇒ 2(-5)/5(5) = dy/dx.
⇒ -10/25 = dy/dx.
⇒ -2/5 = dy/dx.
Put the value of x = 5 in equation, we get.
⇒ 2(5)/5(5) = dy/dx.
⇒ 10/25 = dy/dx.
⇒ 2/5 = dy/dx.
⇒ M₂ = (-2/5,2/5).
As we know that,
⇒ Tan∅ = | m₁ - m₂/1 + m₁.m₂|.
⇒ Tan∅ = 1 - (-2/5)/1 + (1)(-2/5).
⇒ Tan∅ = 1 + 2/5/1 - 2/5.
⇒ Tan∅ = 5 + 2/5/5 - 2/5.
⇒ Tan∅ = 7/5/3/5.
⇒ Tan∅ = 7/3.
⇒ ∅ = tan⁻¹(7/3).
✿...A N S W E R...✿
acute angle between the curves
x²+y²=50 ( 1 )
x²=5y = 2 ( 2 )
put the value of equation ( 1 ) in equation ( 2 )
we get,
➩y² + 5y = 50
➩y² + 5y - 50 = 0
➩y² + 10y - 5y - 50 = 0
➩y ( y + 10 ) - 5 ( y + 10 )
➩ ( y - 5 ) ( y + 10 ) = 0
➩ y = 5 & y = -10
put the value of y in equation
put y = 5 in equation (2) we get ,
⇝x ² = 5 ( 5 )
⇝x² = 25
⇝x = √25
⇝x = + 5
⇝x ² + y ² = 50
➢ 2× + 2y. Dy/dx + y
➢2 x = 5.dy / dx ( y)
➢2× /5y =dy/dx
➢2(5)/5(5) = dy / dx
➢-10/25 = dy / dx
➢2/5 = dy / dx