Question

If x = a(t – sin t) and y = a(1 – cos t), then `("d"y)/("d"x)` = ______.

Options

• `tan ("t"/2)`

• `-tan ("t"/2)`

• `cot ("t"/2)`

• `-cot ("t"/2)`

Answer 1

If x = a(t – sin t) and y = a(1 – cos t), then `("d"y)/("d"x)` = `cot ("t"/2)`.

Explanation:

x = a(t – sin t) and y = a(1 – cos t)

∴ `("d"x)/"dt"` = a(1 – cos t) and `("d"y)/"dt"` = a sin t

∴ `("d"y)/("d"x) = ((dy)/"dt")/((dx)/("dt"))`

= `("a"sin"t")/("a"(1 - cos "t")`

= `(2"a" sin  "t"/2 cos  "t"/2)/(2"a" sin^2  "t"/2)`

= `cot  "t"/2`