Prove that the feet of the perpendiculars from the foci S, S' of an ellipse on a tangent lie on the auxiliary circle. Parallel lines SP, S'P' drawn towards the same parts meet the ellipse in P and P'. Prove that the tangents to the ellipse at P and P' meet on the auxiliary circle.
Prove that the equation \(x^2+y^2-2cx\sec\theta+c^2=0\) as \(\theta\) varies represents a system of coaxal circles with real limiting points L, L'. If P is any point on the circle \(\theta\) prove that PL:PL' = \(\tan\frac{\theta}{2}:1\), if L lies inside the circle.
Find the equation of the tangent at the point \((at^2, 2at)\) on the parabola \(y^2=4ax\). Find the equation of the circle circumscribing the triangle formed by the tangents to the parabola at the points whose parameters are \(t, t', t''\).
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.