Finding the Values of a and b for a Specific Slope on the Curve W

Introduction

The problem presented involves determining specific values of constants a and b such that the curve W defined by the equation x^3xy a - y^2 has a slope of -1/2 at the point (b, b). This problem combines concepts from implicit differentiation and simultaneous equations.

The Equation and Initial Setup

Consider the curve W defined by the equation x^3 x*y a - y^2, where 'a' is a real constant. The goal is to find the values of a and b such that the curve W has a slope of -1/2 at the point (b, b).

Slope Calculation Using Implicit Differentiation

First, we need to find the slope of the curve W at a given point (x, y). To do this, we will use implicit differentiation.

Starting with the equation:

x^3 x*y a - y^2

We differentiate both sides with respect to x:

3x^2 y x*y' -2y*y'

Rearranging to solve for y' (the slope of the curve at a given point), we get:

3x^2 y -2y*y' - x*y'

3x^2 y y'(-2y - x)

y' (3x^2 y) / (2y x)

Applying the Given Slope Condition

At the point (b, b), we have:

y' -1/2

Substitute x b and y b into the slope equation:

-1/2 frac{3b^2 b}{2b b}

-1/2 frac{3b^2 b}{3b}

-1/2 frac{3b 1}{3}

-3/2 3b 1

-3/2 - 1 3b

-5/2 3b

b -5/6

Now, substitute b -5/6 into the equation for the slope to verify:

x^3 x*y a - y^2

(-5/6)^3 (-5/6)*(-5/6) a - (-5/6)^2

(-5/6)^3 (25/6)/6 a - (25/36)

-125/216 25/36 a - 25/36

-125/216 150/216 a - 25/36

25/216 a - 25/36

25/216 a - 75/108

25/216 150/216 a

a 175/216

Therefore, the values of a and b are a 175/216 and b -5/6.

Conclusion

In summary, we have used implicit differentiation to find the slope of the given curve and solved a system of equations to determine the specific values of a and b that satisfy the given conditions.