9. Find the equation of the line intersecting the yaxis at a distance of 2 units above the origin and making an angle of 30° with positive direction of x -axis.
Answers
EXPLANATION.
Equation of the line intersecting at the y- axis at the distance of 2 units.
above the origin.
making an angle of 30° with positive x - axis.
line intersect at y - axis it means the co-ordinate of the point are,
(0,2).
slope of the line = tan ∅ = 30° = 1/√3.
Equation of line. = (y - y₁) = m ( x - x₁)
⇒ ( y - 2 ) = 1/√3 ( x - 0 ).
⇒ √3 ( y - 2 ) = 1 ( x - 0 ).
⇒ √3y - 2√3 = x.
⇒ x - √3y + 2√3 = 0.
MORE INFORMATION.
Test of monotonicity.
Let f(x) is continuous on [ a, b ] and differentiable on (a, b).
(1) = if f(x) is increasing (↑) on [ a, b] then f'(x) ≥ 0 for all x ∈ ( a, b).
(2) = if f(x) is decreasing on [ a , b ] then f'(x) ≤ 0 for all x ∈ ( a, b).
(3) = if f'(x) > 0 for all x ∈ (a, b) then (↑).
(4) = if f'(x) < 0 for all x ∈ ( a, b) then (↓).
(5) = if a function f(x) is defined on (a, b) and f'(x) > 0 for all x ∈ ( a, b)
except for a finite number of points where f'(x) = 0 then f(x) is strictly
increasing (↑) on (a , b).
(6) = if a function f(x) is defined on (a, b) and f'(x) < 0 for all x ∈ (a, b)
except for a finite number of points where f'(x) = 0 then f(x) is strictly
decreasing (↓) on (a, b ).
