The distances of a point from the vertices of an equilateral triangle of unknown size are given. Show how the triangle may be constructed by making first a triangle with the lengths of its sides equal to the three given distances.
A variable triangle \(PQR\) inscribed in a circle has the side \(PQ\) parallel to a fixed chord, and \(QR\) passes through the middle point of the chord. Shew that the side \(RP\) also passes through a fixed point.
An aeroplane has an engine-speed equal to that of the wind in which it is flying, and heads continually for a fixed point at its own level. Shew that it moves along a parabola.
Find the locus of the point of intersection of a variable line through a focus of a conic, and a tangent cutting it at a right angle; and shew that it is a circle touching the conic twice. \par Shew that the same is true if the angle is constant but not a right angle.
Two circles in different planes both touch the line of intersection of the planes at the same point. Shew that if a variable plane touches both the circles, it passes through a fixed point \(O\); and that if \(P\) and \(Q\) are the contact points, the product \(OP.OQ\) is constant.
Shew that for a variable normal to a conic the locus of the middle point of the intercept between the axes is a similar and coaxal conic; and shew that two conics may be mutually related in this way.
A conic has eccentricity \(e\) and focus \((a,b)\); and the corresponding directrix is \(lx+my+n=0\). Write down the equation of the conic, and convert it into a form which exhibits the other focus and directrix.
Ellipses are drawn through the middle points of the sides of the rectangle \((x^2-a^2)(y^2-b^2)=0\). Find the general equation of the family; and shew that they are all cut four times orthogonally by one of the hyperbolas having the diagonals as asymptotes.
Equal circles of radius \(r\) have their centres at the points \((\pm a, 0)\). Shew that tangents drawn to them from any point on the conic \[ r^2(x^2-a^2)+(r^2-2a^2)(y^2-r^2)=0 \] form a harmonic pencil. \par Examine the special cases when the circles (i) touch, (ii) cut orthogonally.
Find the general equation of all pairs of lines having the same angle-bisectors as \(ax^2+2hxy+by^2=0\). \par Shew that the general equation of any conic confocal with \(ax^2+2hxy+by^2+c=0\) may be written in the form \[ (ax^2+2hxy+by^2)+\lambda(x^2+y^2)+c\frac{(a+\lambda)(b+\lambda)-h^2}{ab-h^2}=0. \]