Match the following. (a) The tangent at any point on the curve x = a t 3 , y = a t 4 divides the abscissa of the point of contact in the ratio m : n . Then ∣ m + n ∣ is equal to ( m , n are co-prime)

Question

Match the following.

(a) The tangent at any point on the curve $x = a t^{3}, y = a t^{4}$ divides the abscissa of the point of contact in the ratio $m : n$ . Then $∣ m + n ∣$ is equal to ( $m, n$ are co-prime)	(P) 0
(b) The area of the triangle formed by the normal at the point $(1, 0)$ on the curve $x = e^{s i n y}$ with the co-ordinate axes is	(Q) 1/2
(c) If the angle between the curves $x^{2} y = 1$ and $y = e^{2 (1 - x)}$ at the point $(1, 1)$ is $θ$ , then $tan θ$ is equal to	(R) 4
(d) The length of the subtangent at any point on the curve $y = b e^{x /3}$ is equal to	(S) 3

TestHub · Accepted Answer

(a) d x d y ​ = 3 4 ​ t . Tangent at ( a t 3 , a t 4 ) meets the x-axis at ( 4 a t 3 ​ , 0 ) , which divides the abscissa O → ( a t 3 , 0 ) in the ratio 1 : 3, so ∣ m + n ∣ = 4 (R) . (b) d x d y ​ = x c o s y 1 ​ = 1 at (1, 0); the normal is x + y = 1 , area = 2 1 ​ (Q). (c) Both curves have slope -2 at (1, 1), so θ = 0 and tan θ = 0 (P). (d) Subtangent = ​ d y / d x y ​ ​ = 3 (S).

Mathematics - Application of Derivative Question with Solution | TestHub

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