Question 28 (OR 1st Question) If √(1−𝑥^2 ) + √(1−𝑦^2 ) = a (x − y), then prove that 𝑑𝑦/(𝑑𝑥 ) = √(1 − 𝑦^2 )/√(1 − 𝑥^2 ).
Finding 𝒅𝒚/𝒅𝒙 would be complicated here
To make life easy, we substitute
x = sin A
y = sin B
(As √(1−𝑥^2 )= √(1−sin^2𝐴 )=√(cos^2𝐴 ))
And then solve
Let’s substitute
x = sin A
y = sin B
in our equation
Now
√(1−𝑥^2 ) + √(1−𝑦^2 ) = a (x − y)
Putting x = sin A and y = sin B
√(1−sin^2𝐴 ) + √(1−sin^2𝐵 ) = a (sin A − sin B)
√(cos^2𝐴 ) + √(cos^2𝐵 ) = a (sin A − sin B)
cos A + cos B = a (sin A − sin B)
Using
cos A + cos B = 2 cos (𝐴+𝐵)/2 cos (𝐴−𝐵)/2
and
sin A – sin B = 2 cos (𝐴+𝐵)/2 sin (𝐴−𝐵)/2
2 cos((𝐴 + 𝐵)/2) cos((𝐴 − 𝐵)/2) = a × 2 cos((𝐴 + 𝐵)/2) s𝑖𝑛((𝐴 − 𝐵)/2)
cos((𝐴 − 𝐵)/2) = a s𝑖𝑛((𝐴 − 𝐵)/2)
〖cos 〗((𝐴 − 𝐵)/2)/〖sin 〗((𝐴 − 𝐵)/2) = a
cot((𝐴 − 𝐵)/2) = a
(𝐴 − 𝐵)/2 = 〖𝑐𝑜𝑡〗^(−1) 𝑎
𝐴−𝐵 = 2 〖𝑐𝑜𝑡〗^(−1) 𝑎
Putting back values of A and B
sin^(−1)𝑥−sin^(−1)𝑦 = 2〖𝑐𝑜𝑡〗^(−1) 𝑎
Differentiating w.r.t x
1/√(1 − 𝑥^2 )−1/√(1 − 𝑦^2 )×𝑑𝑦/𝑑𝑥=0
1/√(1 − 𝑥^2 )=1/√(1 − 𝑦^2 )×𝑑𝑦/𝑑𝑥
√(1 − 𝑦^2 )/√(1 − 𝑥^2 )=𝑑𝑦/𝑑𝑥
𝑑𝑦/(𝑑𝑥 ) = √(1 − 𝑦^2 )/√(1 − 𝑥^2 )
Hence proved

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.