Question 87

# If $$x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta and x \sin \theta - y \cos \theta = 0$$, then the value of $$x^2 + y^2$$ equals

Solution

$$x \sin^3 \theta + y \cos^3 \theta = \sin \theta \cos \theta$$ --> eq 1

$$x \sin \theta - y \cos \theta = 0$$

$$x \sin \theta = y \cos \theta$$ --->eq2

substituting in eq1

$$y\cos \theta \sin^2 \theta + y \cos^3 \theta = \sin \theta \cos \theta$$

taking $$y\cos \theta common$$

$$y\cos \theta (\sin^2 \theta + \cos^2 \theta )= \sin \theta \cos \theta$$ { we know $$\sin^2 \theta + \cos^2 \theta = 1$$}

$$y\cos \theta= \sin \theta \cos \theta$$

$$y= \sin \theta$$

substituting in eq 2

$$x \sin \theta = \sin \theta \cos \theta$$

x = $$\cos \theta$$

$$x^2 + y^2$$ =  $$\sin^2\theta + \cos^2\theta$$ = 1