
Let p and q be positive real numbers. Let P denote the point (p,0) and Q denote the point (0,q).
- Show that the equation of the circle C which passes through P, Q and the origin O is x2−px+y2−qy=0.
Substituting in the coordinates (p,0), (0,q) and (0,0), we find that all three points satisfy the equation and so lie on the circle, so we are done.
Alternatively, we can deduce the equation from scratch once we know the centre of the circle.
The line PQ is a diameter of C, since the angle POQ is a right-angle, and we know that the angle in a semi-circle is a right-angle.
So the midpoint of PQ, which is M(p2,q2), is the centre of the circle. Thus the circle has equation (x−p2)2+(y−q2)2=r2, for some r. We can find r either by substituting the coordinates of a point on C, say (x,y)=(0,0), which gives r2=(p2)2+(q2)2=p2+q24, or by finding the radius, which is OM=AM=BM, giving the same result.
Find the centre and area of C.
Just using the given equation, we can complete the square to obtain (x−p2)2+(y−q2)2−p2+q24=0 which is the equation of the circle with centre (p2,q2) and area πp2+q24.
Alternatively, using the argument above, we already know the centre of the circle and r2, and the area is πr2=πp2+q24.
- Show that area of circle Carea of triangle OPQ≥π.
The area of OPQ is 12pq, so area of circle Carea of triangle OPQ=π(p2+q2)/4pq/2=πp2+q22pq
Then we note that πp2+q22pq≥π⟺p2+q2≥2pq⟺(p−q)2≥0 which always holds, and so the required inequality is true.
- Find the angles OPQ and OQP if area of circle Carea of triangle OPQ=2π.
From our previous expression we have πp2+q22pq=2π⟺p2+q2=4pq.
Then if we divide through by q2 (which we know is non-zero), we get (pq)2−4(pq)+1=0 which is a quadratic in pq with solution pq=4±√16−42=2±√3.
Now since triangle OPQ is right-angled, we have that pq=tanOQP and qp=tanOPQ.
Then since the solutions are reciprocals of each other, we have {tanOQP,tanOPQ}={2+√3,2−√3}, where the order depends on which of p and q is larger: if p>q, then tanOQP=2+√3; if p<q, then tanOQP=2−√3.
Thus {OQP,OPQ}={arctan(2+√3),arctan(2−√3} in some order, depending on whether p>q or p<q.
[Incidentally, it turns out that arctan(2−√3)=π12 and arctan(2+√3)=5π12, as we will learn at Trigonometry: Compound Angles.]