Prove that four normals can be drawn from a given point to the ellipse \[ x^2/a^2+y^2/b^2=1 \] and prove that the feet of the normals lie on a rectangular hyperbola. If P, Q, R, S, the feet of the four normals, are such that PQ passes through a focus, prove that RS passes through the foot of the directrix corresponding to the other focus. Also shew that PQ and RS intersect on the hyperbola \(x^2/a^2-y^2/b^2+x(1-e^2)/ae=1\).
Find the equation of all conics confocal with
S is a focus of a conic, and the tangent at P meets the corresponding directrix in R, and the corresponding latus rectum in U. Prove that PSR is a right angle, and that \(\frac{SU}{SR}\) is equal to the eccentricity.
Prove that the limiting points of a system of coaxal circles are inverse points with respect to every circle of the system. Prove also that a common tangent to two circles of the system subtends a right angle at either limiting point.
A fly is crawling from one corner of a rectangular matchbox, the lengths of whose edges are a, b and c (\(a>b>c\)), to the opposite (i.e. most distant) corner, round any of the sides of the box. Prove that its shortest possible path is of length \[ \sqrt{a^2+b^2+c^2+2bc}. \]
Prove that any two lines in space are cut proportionately by three parallel planes. AB is the common perpendicular of two straight lines AP and BQ in space; H is the middle point of AB, and M the middle point of PQ. Prove that HM is perpendicular to AB.
Tangent lines are drawn to a sphere from a given external point. Prove that the points of contact lie on a circle. A sphere of radius 6 inches rests on three horizontal wires forming a plane triangle whose sides are 5 inches, 12 inches and 13 inches. Find the height of the top of the sphere above the plane of the wires.
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 \(\frac{1}{2}(a^2-b^2)\).