Find the equation of the straight lines that bisect the angles between the straight lines \[ ax^2+2hxy+by^2=0. \] Prove that the line \[ lx+my+n=0 \] forms with the lines \[ (lx+my)^2=3(mx-ly)^2 \] an equilateral triangle.
Prove that the line \(ty=x+at^2\) touches the parabola \(y^2=4ax\), and find the co-ordinates of the point of contact. Prove that the locus of the centre of a circle which passes through the vertex of the parabola and touches it at some other point is \[ 2(x-2a)^3 = 27ay^2. \]
Give a definition of the polar of a point \((h,k)\) with respect to the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \] which will apply when the point is inside the ellipse, and find the equation of the polar from the definition. Obtain the equations of a pair of lines at right angles to each other and such that each passes through the pole of the other. Prove that the product of the distances of such a pair of lines from the centre depends only on their direction, and cannot exceed \(\frac{1}{2}(a^2-b^2)\).
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.