Find the The Slopes of the tangent and the normal to the following curves at the indicated points :
x = a (θ – sin θ), y = a(1 + cos θ) at
θ = – π/2
Given:
x = a() & y = a(1 + cos) at
Here, To find , we have to find & and and divide and we get our desired .
(xn) = n.xn – 1
⇒ x = a()
⇒ = a(() – (sin))
⇒ = a(1 – ) ...(1)
(sinx) = cosx
⇒ y = a(1 + cos)
⇒ = a(() + (cos))
(cosx) = – sinx
(Constant) = 0
⇒ = a( + ( – sin))
⇒ = a( – sin)
⇒ = – asin ...(2)
⇒
⇒
The Slope of the tangent is
Since,
⇒
sin() = 1
cos() = 0
⇒
⇒
⇒ = 1
The Slope of the tangent at x = is 1
⇒ The Slope of the normal =
⇒ The Slope of the normal =
⇒ The Slope of the normal =
⇒ The Slope of the normal = – 1