Question

Find the highest and lowest points on the curve x² + xy + y² = 12

Answer 1

We have x² + xy + y² = 12

Put x = r cosθ, y = rsinθ

⇒ r²cos²θ + r²sinθcosθ + r²sin²θ = 12

⇒ r² + r²sinθcosθ = 12

⇒ r²(1 + sinθcosθ) = 12

i.e. r = 2√6

and (x, y) = (r cosθ, rsinθ)

= (2√3 , – 2√3)

Min value of r 2 is 24/3 = 8

i.e r = 2√2 and (x, y) = (2, –2)