Vertical tangents on a tilted ellipse

The equation of a closed curve is given by (nx+y)2+4(nx−y)2=20(nx+y)^2+4(nx-y)^2=20, where nn is a constant. Find

(i) the gradient of the curve at the point (x,y)(x,y), marks

(ii) the coordinates of the points on the curve at which the tangent is parallel to the yy-axis, marks

(iii) the equations of the normal at the points where the curve cuts the xx-axis. marks

[2009/IJC/Prelim/I/11]

Total marks: 10

Solution

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  1. (i) Implicit differentiation →
  2. (ii) Vertical tangents →
  3. (iii) Normals →