Moderate -0.3 This is a straightforward implicit differentiation question requiring students to differentiate both sides with respect to x, substitute the given point to find the gradient, then write the tangent equation. While it involves multiple steps (differentiate, solve for dy/dx, substitute coordinates, form equation), each step is routine and follows a standard algorithm with no conceptual challenges or novel insights required. Slightly easier than average due to the mechanical nature of the process.
5 A curve has equation \(x ^ { 2 } + 2 y ^ { 2 } + 5 x + 6 y = 10\). Find the equation of the tangent to the curve at the point \(( 2 , - 1 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. [0pt]
[6]
Substitute \(2\) and \(-1\) to attempt value of \(\frac{dy}{dx}\)
M1
Obtain \(-\frac{9}{2}\)
A1
Obtain equation \(9x + 2y - 16 = 0\) or equivalent of required form
A1
[6]
Obtain $4y\frac{dy}{dx}$ as derivative of $2y^2$ | B1 |
Differentiate LHS term by term to obtain expression including at least one $\frac{dy}{dx}$ | M1 |
Obtain $2x + 4y\frac{dy}{dx} + 5 + 6\frac{dy}{dx}$ | A1 |
Substitute $2$ and $-1$ to attempt value of $\frac{dy}{dx}$ | M1 |
Obtain $-\frac{9}{2}$ | A1 |
Obtain equation $9x + 2y - 16 = 0$ or equivalent of required form | A1 | [6]
5 A curve has equation $x ^ { 2 } + 2 y ^ { 2 } + 5 x + 6 y = 10$. Find the equation of the tangent to the curve at the point $( 2 , - 1 )$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.\\[0pt]
[6]
\hfill \mbox{\textit{CAIE P2 2011 Q5 [6]}}