If the normal at \(P\) to a hyperbola meet the axes in \(G\) and \(g\), prove that the ratio \(PG:Pg\) is constant. \par Prove that the normal and tangent to a hyperbola at any point meet the axes and asymptotes respectively in four points which lie on a circle through the centre of the conic.
Prove that the polar reciprocal of a circle with regard to another circle is a conic section. \par \(AB, AC, BC\) are three tangents to a conic the point \(A\) being on another conic having the same focus and directrix. Shew that the angle that \(BC\) subtends at the common focus is constant.
Explain the method of proving propositions by projection, stating what classes of properties are projective and illustrating by examples.
Prove that for different values of \(p\) the centroid of the triangle whose sides are \[ x\cos\alpha+y\sin\alpha-p=0 \] and \[ ax^2+2hxy+by^2=0 \] lies on the line \[ x(a\tan\alpha-h)+y(h\tan\alpha-b)=0. \]
Find the equation of the normal at any point on the curve \[ x=am^2, \quad y=2am. \] Shew that the normals to the curve at the extremities of the chords \[ y=4c, \quad ax+cy+\kappa^2=0 \] are concurrent.
Find equations for determining the foci of the conic represented by the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Find the foci of the curve \[ 3x^2+3y^2-2xy-8x+8y-24=0, \] and draw the curve.
If \(S=0\) be the equation of a circle and \(\alpha=0, \beta=0\) are the equations of straight lines, assign geometrical meanings to the expressions \(S, \alpha, \beta\). \par Also interpret the equations \[ S-\kappa\alpha\beta=0, \quad S-\kappa\alpha^2=0 \] and state what geometrical theorems they represent.
Find the condition that the four lines given by the equations \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] may represent a harmonic pencil. \par Shew that the same condition would make the first pair of lines conjugate diameters of the conic \[ a'x^2+2h'xy+b'y^2=1. \] Explain the relation of these results in reference to the theory of pencils in involution.
If a circle \(S\) touches the circumscribed circle of a triangle \(ABC\) at \(P\), prove that the tangents to \(S\) from \(A, B, C\) are in the ratio \(AP:BP:CP\). \par What does the result become when the radius of the circle \(S\) tends to infinity?
Prove that if in a plane the ratio of the distances from two points are the same for each of the three points \(A, B, C\), the two points are inverse points with regard to the circle \(ABC\). \par Shew that the line bisecting \(BC\) at right angles meets the lines \(BA\) and \(CA\) in two inverse points.