Given two intersecting straight lines and a point in a plane, shew how to draw the straight line, the intercept on which between the given lines is bisected at the point, and also the two lines for which the intercepts are trisected.
In a plane a circle is given and two points external to it. Shew how to construct the two circles which pass through the given points and touch the given circle.
In a parabola \(SY\) is the perpendicular from the focus \(S\) on the tangent at the point \(P\) and \(A\) is the vertex, prove that \(SY^2 = SA.SP\). Prove that, if \(PM, PN\) be the perpendiculars on the tangent at the vertex and on the axis respectively, \(MN\) touches a parabola.
Prove that, if \(A\) and \(B\) two points on a conic be each joined to four given points on the conic, the resulting pencils of four lines have the same cross ratio. Two points \(S\) and \(H\) are the foci of a variable conic inscribed in a triangle \(ABC\): shew that, if \(S\) describes a straight line, \(H\) describes a conic circumscribing the triangle \(ABC\).
Prove that through two circles which are plane sections of the same sphere it is possible to construct two cones and that the line joining their vertices is the polar line with regard to the sphere of the line of intersection of the two planes.
Prove that \(x = \mu^2 - \lambda^2, y=2\lambda\mu\) is a point of intersection of the two confocal parabolas of the systems obtained by making in turn the two parameters \(\lambda\) and \(\mu\) the one constant and the other variable. Shew also that the two parabolas cut at right angles.
Prove that the equation of the line of a chord of the ellipse \(x^2/a^2+y^2/b^2=1\) may be written \[ \frac{x}{a} t_1 t_2 \left(\frac{a}{x}+1\right) - \frac{y}{b}(t_1+t_2) + \left(1-\frac{x}{a}\right) = 0, \] where \(t_1, t_2\) denote the tangents of the halves of the eccentric angles of the ends of the chord. A point \(Q\) on the auxiliary circle of an ellipse is joined to the extremities \(A, A'\) of the major axis, the joining lines cutting the ellipse again in \(P\) and \(P'\) respectively: shew that the line \(PP'\) envelopes an ellipse.
Shew that the condition that \(lx+my-1=0\) shall be normal to the ellipse \(x^2/a^2+y^2/b^2=1\) is \[ a^2m^2+b^2l^2 = (a^2-b^2)^2 l^2 m^2. \] Shew that two coaxal and concentric conics have four common normals in addition to the axes and prove that they are real lines for the conics \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}-1=0 \quad \text{and} \quad \frac{x^2}{a^2-\mu}+\frac{y^2}{b^2+\mu}-1=0, \] provided \[ \frac{(a^2-b^2)(a^2+3b^2)}{b^2} > 4\mu > \frac{(a^2-b^2)(3a^2+b^2)}{a^2}. \]
A family of conics have their centres at the origin and the lines \(x=\pm d\) as directrices: prove that two members of the family, which are both ellipses or both hyperbolas, pass through a chosen point in the plane provided the point lie in the regions surrounding the axis of \(x\) and bounded by the two parabolas \[ x^2-d^2-2dy=0, \quad x^2-d^2+2dy=0. \] Prove also that any member of the family touches both parabolas but that, when the member is an ellipse, the points of contact are not real if the eccentricity is less than \(1/\sqrt{2}\).
Prove that the circle of curvature at the point \((am^2, 2am)\) on the parabola \(y^2-4ax=0\) is given by the equation
\[
x^2+y^2-ax(4+6m^2)+4aym^3-3a^2m^4=0
\]
and that there are four circles of curvature real or imaginary through a chosen point.
In the case of a point \((x, 0)\), shew that no real circle of curvature passes through the point if \(0