By P.S. MacIlwaine, Charles Plumpton, P. S. W. MacIlwaine

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22 Given that a > k > b > 0, prove that the ellipse x 2 1a 2 + y 21b 2 = I cuts the hyperbola in four real points . If P is anyone of these points , prove that the tangents at P to the two curves are perpendicular. Find an equation of the circle which passes through the four points. 23 Tangents are drawn to the rectangular hyperbola xy = c 2 at the points P(ct I ' cit I) and Q(ct 2 , clt 2 ) . Find the coordinates of the orthocentre of the triangle formed by these tangents and the line y = O. Given that P and Q vary so that the line PQ passes through a fixed point on the line y = 0, show that the orthocentre lies on a fixed parabola.

Given that P and Q vary so that the line PQ passes through a fixed point on the line y = 0, show that the orthocentre lies on a fixed parabola. 24 Given that the normal to the hyperbola xy = c 2 at the point x = ct, Y = cit passes through the point P(h , k), show that ct" - ht 3 + kt - c = O. The normals at four points on the hyperbola meet at P. Show that the sum of the x-coordinates of the four points is h, and that the sum of their y-coordinates is k. 25 S is the parabola with parametric equations x = at", y = 2at, and PI and P2 are the points of S with parameters t l and t 2 respectively.

Iii) The polar equation r = a, where a > 0, represents the circle with centre the pole and radius a. If r = f(cos e), since cos (-e) = cos e, the curve is symmetrical about the initial line. If r = g(sin e), since sin (tn - e) = sin (tn + e), the curve is symmetrical about the half-lines e = ±tn. To obtain a curve with polar equation r = h(cos e, sin e), clearly we need only consider e in the range ~ e < 2n (or -n < e ~ z). Considerations of symmetry together with some convenient values of e will usually enable us to sketch a polar curve.