Question

Let Cbe a curve passing through M (2, 2) such that the slope of the tangent at any point to the curve
is reciprocal of the ordinate of that point. If the area bounded by curve Cand line x = 2 is expressed as
a rational number (where p and q are in their lowest form), then value of (p+q) is greater than
prime numbers
(B) 11
(A) 5
(D) 19
(C) 17​

Answer 1

Given : C is  a curve passing through M (2, 2) such that the slope of the tangent at any point to the curve  is reciprocal of the ordinate of that point.

the area bounded by curve Cand line x = 2 is expressed as  a rational number (where p and q are in their lowest form)

To find :  value of (p+q) is greater than  prime numbers

Solution:

slope of the tangent at any point to the curve  is reciprocal of the ordinate of that point.

=>  dy/dx  = 1/y

=> ydy = dx

integrating both sides

=> y²/2  = x  + c

=> y² = 2x  + c

curve passing through M (2, 2)

=> 2² = 2(2) + c

=> c = 0

Hence y² = 2x  

curve C    y² = 2x    =>y = √2√x

x = 0  to 2

A = $2\times \int\limits^2_0 {\sqrt{x} } \, dx$

Area = $2\times [ {\frac{2x\sqrt{x} }{3}} ]_0^2$

=  16/3

p = 16

q = 3

p + q = 16 + 3 = 19

Greater than 17

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