Can we show that these normals intersect on this circle?

Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Prove that the equation of the normal at a point of the curve \[x=n\cos t-\cos nt,\qquad y=n\sin t-\sin nt,\] where \(n\) is an integer greater than unity, is \[x\cos \frac{n+1}{2}t + y\sin \frac{n+1}{2}t=(n-1)\cos \frac{n-1}{2}t.\] Show that, if \(n\) is an even integer, the normals at the points \(t\) and \(t+\pi\) are perpendicular and intersect on the circle \[x^2+y^2=(n-1)^2.\]