This is an excellent Edexcel exam question from the International Further Pure 3 (June 2024) paper. I like it because it connects a wide range of skills — from using double angle formulae in pure mathematics to applying the vector cross product from Further Pure 1.
We start by finding the general equation of the normal to an ellipse. Using the chain rule, we calculate the gradient of the tangent at a general point, then use this gradient and point to form the equation of the normal.
In the second part, we look at the locus of the midpoint of a line segment formed by the normal. After finding the coordinates of the midpoint, we use the vector cross product to calculate the area of the triangle formed with the origin. Finally, we express the area in terms of a double angle to determine its maximum value.
The full lesson on locus of conic sections includes:
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