Defining a diameter of a parabola as the locus of the middle points of parallel chords, prove that a diameter is a straight line parallel to the axis. The diameter through any point \(O\) of a parabola meets any chord \(PQ\) in \(R\), the tangent at \(P\) in \(T\), and the ordinates through \(P, Q\) to the diameter in \(V, V'\). Prove that \[ OR^2 = OV \cdot OV', \text{ and } \frac{TO}{OR} = \frac{PR}{RQ}. \]
Establish the harmonic property of the complete quadrilateral. Given two parallel straight lines \(AB\) and \(CD\), bisect \(AB\) and \(CD\) by means of a ruler only.
Prove that if two planes are each perpendicular to a third plane, their line of intersection is perpendicular to that plane. Prove that, if the perpendiculars from two vertices of a tetrahedron on the opposite faces intersect, then the perpendiculars from the other two vertices on the opposite faces also intersect.
Find the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] should represent a pair of straight lines; and if it does, and if also \[ (a-b)fg+h(f^2-g^2)=0, \] prove that these straight lines form a rhombus with the straight lines \[ ax^2+2hxy+by^2=0. \]
Prove that the square of the tangent from a point to the circle \[ x^2+y^2+2gx+2fy+c=0 \] is obtained by substituting the coordinates of the point in the equation of the circle. Points \(P\) and \(P'\) are taken, the first on a given circle \(S\), and the second on a given circle \(S'\). If the tangent from \(P\) to \(S'\) is equal to that from \(P'\) to \(S\), show that \(P, P'\) are equidistant from the radical axis of \(S, S'\).
Prove that the equation of the chord of the parabola \[ y^2=4ax \] whose middle point is \((x',y')\) is \[ yy' - 2ax = y'^2 - 2ax'. \] Find the envelope of chords of the parabola whose middle points are on the given line \[ y=mx+c. \]
Find the condition that the line \[ \frac{x}{p}+\frac{y}{q}=1 \] may be a normal to the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1; \] and prove that in general there is one conic, and one only, with given centre and directions of axes, which has two given lines as normals.
Obtain the equation of a hyperbola referred to its asymptotes as (oblique) axes in the form \[ 4xy = a^2+b^2. \] A given hyperbola is cut by any circle in the points \(P, Q, R, S\). Prove that the product of the perpendiculars from \(P, Q, R, S\) on an asymptote is the same for all circles.
Find the conditions that \(ax^2+2bx+c\) may keep one sign for all real values of \(x\). Shew that if \(x,y\) are real, and \(x^2+xy+y^2=1\), then \(x+2y\) must lie between \(-2\) and \(+2\).
Find the factors of \[ a^3(b-c)+b^3(c-a)+c^3(a-b). \] Shew that if \begin{align*} x^3+y^3+z^3 &= 3xyz, \\ ax^2+by^2+cz^2 &= 0, \\ ayz+bzx+cxy &= 0, \end{align*} then \[ a^3+b^3+c^3 = 3abc. \]