Question

The equation of tangent to the curve `(x/"a")^"n" + (y/"b")^"n"` = 2 at the point (a, b) is ______.

Options

• `x/"a" = -y/"b"`

• `x/"a" + y/"b"` = 2

• `x/"a" = y/"b"`

• `x/"a" + y/"b"` = n

Answer 1

The equation of tangent to the curve `(x/"a")^"n" + (y/"b")^"n"` = 2 at the point (a, b) is `x/"a" + y/"b"` = 2.

Explanation:

`(x/"a")^"n" + (y/"b")^"n"` = 2

Differentiating w.r.t x, we get

`"n"(x/"a")^("n" - 1) (1/"a") + "n"(y/"b")^("n" -1) (1/"b")(("d"y)/("d"x))` = 0

⇒ `"n"/"b" (y/"b")^("n" - 1) ("d"y)/("d"x) = (-"n")/"a" (x/"a")^("n" - 1)`

⇒ `("d"y)/("d"x) = (-"b")/"a" (x/"a")^("n" - 1) ("b"/y)^("n" - 1)`

Slope of tangent at (a, b) = `(("d"y)/("d"x))_((("a", "b")))`

= `(-"b")/"a" ("a"/"a")^("n" - 1) ("b"/"b")^("n" - 1)`

= `(-"b")/"a"`

Equation of tangent is `(y - "b") = (-"b")/"a"(x - "a")`

⇒ ay – ab = –bx + ab

⇒ ay + bx = 2ab

⇒ `x/"a" + y/"b"` = 2