If ( a + √2 b cos x )( a - √2 b cos y ) = a² - b²
where a > b > 0, then dx/dy at( π/4,π/4 ) is :

Solution

Maths sir · 4 steps

Differentiate with respect to y, use the product rule, and at π/4 every √2 trig factor becomes 1.

1. Product rule in y. The right side a² - b² is a constant, so its derivative is 0. Differentiate both brackets with respect to y, treating x as a function of y, so cos x gives -sin x dx/dy.

   (-√2 b sin x dx/dy)(a - √2 b cos y) + (a + √2 b cos x)(√2 b sin y) = 0

2. Put in the point. At x = y = π/4, sin and cos are both 1/√2, so each of √2 b sin x, √2 b cos y, √2 b cos x and √2 b sin y equals b. The equation becomes -b dx/dy(a - b) + (a + b) b = 0.
3. Solve. Divide by b (it is not zero) and rearrange: dx/dy(a - b) = a + b. Since a > b, a - b ≠ 0, so we can divide by it.
4. So. dx/dy = (a + b)/(a - b), which is option (c).

The idea to keep. In implicit differentiation, first note which variable you differentiate with respect to. Substitute the point right after differentiating, before you rearrange. The algebra shrinks a lot.

Answer from the NTA final answer key. Solution by PrepWiser.