Prove that a tangent of an ellipse is equally inclined to the focal radii of its point of contact. If \(TP, TQ\) be tangents, at \(P\) and \(Q\), of an ellipse whose foci are \(S\) and \(H\), and \(SP, HQ\) intersect in \(R\), prove that the angles \(TRP, TRQ\) are equal.
Shew that the area included between a tangent to a hyperbola and the two asymptotes is constant. A variable chord of a circle is divided harmonically by two fixed lines through the centre: prove that the envelope of the chord is a hyperbola with the lines as asymptotes.
Shew that the locus of the intersection of tangents to a parabola which meet at a constant angle is a conic having double contact with the parabola along its directrix. Reciprocate this theorem with respect to any point.
Shew that the focus of a conic divides any chord through it so that the rectangle contained by the parts is proportional to the length of the chord, and shew also that the foci are the only points which possess this property.
Find the equation of the circle of which the chord of the ellipse \[ ax^2+by^2=1 \] intercepted on the line \(lx+my=1\) is a diameter. Prove that this circle will touch the ellipse at a third point if the given chord touches a similar, similarly situated and concentric ellipse.
Prove that the conic \[ 9x^2 - 24xy+41y^2 = 15x+5y \] has one extremity of its major axis at the origin and one extremity of its minor axis on the axis of \(x\). Find the coordinates of its centre and foci.
Shew that the locus of the point whose homogeneous coordinates \(x,y,z\) are given in terms of a parameter \(t\) by the equations \[ x/(at^2+2a't+a'')=y/(bt^2+2b't+b'')=z/(ct^2+2c't+c'') \] is a conic whose tangential equation is \[ (al+bm+cn)(a''l+b''m+c''n)-(a'l+b'm+c'n)^2=0. \] Shew that, if two points are given by parameters \(t,t'\) related by the condition \(Att'+B(t+t')+C=0\), the chord joining them passes through a fixed point.
A line \(MN\) of given length slides between two fixed straight lines \(OX, OY\), \(M\) being on \(OX\) and \(N\) on \(OY\). \(MP, NP\) are perpendicular to \(OM, ON\). Prove that the locus of \(P\) is a circle.
The sides of a triangle \(ABC\) are cut by a straight line in \(D, E, F\). Prove that \[ BD \cdot CE \cdot AF = -CD \cdot AE \cdot BF. \] If \(D'\) is the harmonic conjugate of \(D\) with respect to \(B,C\); and if \(E', F'\) are similarly determined on \(CA, AB\), shew that \(AD', BE', CF'\) are concurrent at a point \(O\). If the straight line \(DEF\) is parallel to one side of the triangle, shew that \(O\) lies on the corresponding median.
\(TP, TQ\) are tangents to a parabola. Prove that the angle \(TP\) makes with the axis is equal to the angle \(TQ\) makes with the focal distance of \(T\). Prove that the circle circumscribing the triangle formed by three tangents to a parabola passes through the focus.