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The curve C has equation $$x^2 - 3xy - 4y^2 + 64 = 0$$ (a) Find $\frac{dy}{dx}$ in terms of x and y - Edexcel - A-Level Maths: Pure - Question 3 - 2015 - Paper 4
Question 3
The curve C has equation $$x^2 - 3xy - 4y^2 + 64 = 0$$ (a) Find $\frac{dy}{dx}$ in terms of x and y. (b) Find the coordinates of the points on C where $\frac{dy}{... show full transcript
Worked Solution & Example Answer:The curve C has equation $$x^2 - 3xy - 4y^2 + 64 = 0$$ (a) Find $\frac{dy}{dx}$ in terms of x and y - Edexcel - A-Level Maths: Pure - Question 3 - 2015 - Paper 4
Step 1
Find $\frac{dy}{dx}$ in terms of x and y.
Answer
To find , we will perform implicit differentiation on the equation of the curve:
Starting from:
Differentiating both sides with respect to x:
This leads to:
Expanding and collecting terms gives:
Rearranging the equation:
Thus, we can solve for :
Step 2
Find the coordinates of the points on C where $\frac{dy}{dx} = 0$.
Answer
Setting the numerator of equal to zero gives:
From here, we can express y in terms of x:
We will substitute this into the original equation to find the coordinates:
Substituting into:
We have:
Simplifying this:
This gives:
Combining the x terms:
Solving for x gives:
Now substituting back to find y:
When :
When :
Thus, the points are: