Answer:
The value of a = ±(√3)/(6)
Step-by-step explanation:
Points A and D belong to the x−axis.
All vertices on the parabola y = a (x+1)(x−5) = a (x² - 4x - 5)
So, points A and D represents the x-intercept of the parabola y
To find x-intercept, put y = 0
∴ a (x+1)(x−5) = 0 ⇒ divide both sides by a
∴ (x+1)(x−5) = 0 ⇒ x = -1 or x = 5
so, the x-coordinate of Point A is -1 or 5
And given that: m∠BAD=60°
So, the tangential line of the parabola at point A has a slope of 60°
∴ y' = tan 60° = √3
∴ y' = a (2x-4)
∴ a (2x-4) = √3
∴ a = (√3)/(2x-4)
Substitute with x = -1 ⇒ a = (√3)/(-6)
Substitute with x = 5 ⇒ a = (√3)/(6)
So, The value of a = ±(√3)/(6)
Also, see the attached figure, it represents the problem in case of a = (√3)/(-6)
