Additional Mathematics – Calculus | e-Consult

Answer 3

Differentiate both sides with respect to \(x\):

\(2x + y + x\frac{dy}{dx} + 2y\frac{dy}{dx}=0.\)

Collect \(\frac{dy}{dx}\) terms:

\((x+2y)\frac{dy}{dx}= - (2x + y).\)

Hence

\(\displaystyle \frac{dy}{dx}= -\frac{2x+y}{x+2y}.\)

At \((1,2)\):

\(\displaystyle \left.\frac{dy}{dx}\right|_{(1,2)}= -\frac{2(1)+2}{1+2(2)}= -\frac{4}{5}.\)

The gradient of the tangent is \(-\frac{4}{5}\). The point is \((1,2)\). Using point‑slope form:

\(y-2 = -\frac{4}{5}(x-1).\)

Multiplying through by 5 gives the tidy form:

\(5y-10 = -4x+4\) → \(4x+5y=14.\)