Problems

Sketch the curve given by the equation \[ y^2 = \frac{x^2(3a-x)}{a+x}. \] Shew that the coordinates of any point on the curve may be taken as \((a\sin 3\theta\csc\theta, a\sin 3\theta\sec\theta)\), and prove that the area of the loop of the curve is equal to the area between the curve and its asymptote.

Prove that there are in general two points \(P\) in the plane of a triangle \(ABC\), such that \(PA:PB:PC = \alpha:\beta:\gamma\), where \(\alpha, \beta, \gamma\) are given numbers. Discuss the position of the point \(P\), when \(\alpha = \beta = \gamma\).

Two coplanar curves are inverted with respect to a point in their plane; prove that the inverse curves cut at the same angle as the original curves. If \(t\) is the length of a common tangent of two coplanar intersecting circles of radii \(a, b\), prove that the ratio \(t^2:ab\) is unaltered by inversion.

\(O\) is a fixed point and \(P\) a variable point on a fixed line; find the envelope of the line through \(P\) perpendicular to \(OP\). A variable ellipse has a given focus and touches two given lines; prove that the envelope of its minor axis is a parabola.

Prove that the polar reciprocal of a circle with respect to a coplanar circle is a conic, and determine the lines which reciprocate into the centre and the foci of this conic. \(T, T'\) are a pair of conjugate points on the directrix of a rectangular hyperbola and tangents from \(T, T'\) touch the rectangular hyperbola at \(P, P'\). Prove that \(PP'\) touches a parabola whose directrix is \(TT'\).

The points of a circle lying in a plane \(p\) are joined to a point external to \(p\); prove that the cone so formed is cut by any plane parallel to \(p\) in a circle, and that there is in general another system of parallel planes which cut the cone in circles. What is the exceptional case?

(i) Prove that one diagonal of the parallelogram formed by the lines \[ ax+by+c=0, \quad ax+by+c'=0, \quad px+qy+r=0, \quad px+qy+r'=0 \] has for its equation \[ (r-r')(ax+by+c) = (c-c')(px+qy+r). \] (ii) Prove that the reflexion of the line \(a'x+b'y+c'=0\) in the line \(ax+by+c=0\) has for its equation \[ 2(aa'+bb')(ax+by+c) - (a^2+b^2)(a'x+b'y+c')=0. \]

Find the coordinates of the centres of similitude of the circles \[ x^2+y^2-2ax=0, \quad x^2+y^2-2by=0. \] Hence, or otherwise, prove that the equations of the two real common tangents of these circles are \[ ax+by+ab \pm \sqrt{2ab}(x+y)=0. \]

If \(lx+my+n=0\) is the tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0\) at a point whose normal passes through the point \((x_1, y_1)\), prove that \[ (a^2-b^2)lm+x_1mn-y_1nl=0. \] Hence prove that the tangents at the feet of the four normals through any point \(P\) touch a parabola, whose directrix is the diameter through \(P\) of the ellipse.

Determine the \((x,y)\) equation of all conics confocal with the conic \[ 3x^2+4xy-4=0,\] and find the equations of the two conics of the system which pass through (i) the point \((4,2)\), and (ii) the point \((2,1)\). Interpret these results.