Prove that if the conics \(S=0, S'=0\) have a pair of common chords \(\alpha=0, \beta=0\) such that \(S-S'= \alpha\beta\), the equation \[ k^2\alpha^2-2k(S+S')+\beta^2=0 \] represents a conic having double contact with each of the conics S and S'. Prove that the conic \[ c(x^2+y^2)+2xy\sqrt{(a-c)(b-c)}=1 \] has double contact with each of the conics \[ ax^2+by^2=1 \] and \[ bx^2+ay^2=1. \]
If \(e\) is the eccentricity of the conic \[ ax^2+2hxy+by^2=1, \] prove that \[ \frac{e^4}{1-e^2} = \frac{(a-b)^2+4h^2}{ab-h^2}. \] A chord PQ of a circle subtends an angle \(2\alpha\) at the centre, and O is the middle point of PQ. An ellipse whose centre is O cuts the circle at P and Q and touches it at a third point. Prove that the eccentricity of the ellipse is \[ \frac{\sqrt{\cos\alpha}}{\cos\frac{\alpha}{2}}. \]
Prove that the external bisectors of the angles of a triangle meet the opposite sides in collinear points.
Prove that the circle drawn through the middle points of the sides of a triangle also passes through the feet of the perpendiculars from the angular points on the opposite sides.
Prove that, if the polar of a point P with respect to a circle pass through the point Q, the polar of Q will pass through P. Prove also that the circle described on PQ as diameter will cut the given circle orthogonally.
Prove that the inverse of a circle is a circle or a straight line, and that, if it is a straight line, that line is the radical axis of the given circle and the circle of inversion.
Prove that the focus of a parabola which touches the sides of a triangle lies on the circumscribing circle of the triangle. Shew that, if ABC is the triangle and P any point on the circumscribing circle, the tangents from P are equally inclined to BC and AP.
Prove that if CP, CD are conjugate semi-diameters of an ellipse whose foci are S and S', the rectangle SP.S'P is equal to the square on CD. Prove also that if SP and CD intersect in E, then \(CP^2-SE^2\) is equal to the square on the semi-minor axis.
Solve the equations \[ x(y+a)-ay = y(z+a)-az = z(x+a)-ax \] \[ 3(x+y+z)=10a. \] Resolve into partial fractions \[ \frac{1}{(1-2x)(1-8x^3)}. \]
Assuming that the series \[ 1+6x+12x^2+kx^3+120x^4+408x^5+\dots \] is a recurring series, determine the value which k must have and find the general term of the series.