10273 problems found
\(PQ\) is any chord of a parabola. Any line parallel to the axis of the parabola meets \(PQ\) in \(E\), the curve in \(R\), and the tangent at \(P\) in \(F\). Prove that \(\dfrac{FR}{EP} = \dfrac{ER}{EQ}\).
The tangent to an ellipse at any point \(P\) meets a given tangent in \(T\). From a focus \(S\) a line is drawn perpendicular to \(ST\), meeting the tangent at \(P\) in \(Q\). Prove that the locus of \(Q\) is a straight line that touches the ellipse.
\(S\) and \(H\) are the foci of a hyperbola. The tangent at \(P\) meets an asymptote in \(T\). Prove that the angle between that asymptote and \(HP\) is twice the angle \(STP\).
Four equal spheres of radius \(r\) all touch one another. Find the radius of the smallest sphere that could enclose them all.
Prove that the two straight lines \[ x^2 \sin^2\alpha \cos^2\theta + 4xy \sin\alpha \sin\theta + y^2\{4\cos\alpha-(1+\cos\alpha)^2\cos^2\theta\} = 0 \] meet at an angle \(\alpha\).
An ellipse of given eccentricity \(\sin 2\beta\) passes through the focus of the parabola \(y^2 = 4ax\) and has its own foci on the parabola. Prove that the major axes of all such ellipses touch the parabola \[ y^2 = 4a(1-\tan^2\beta)(x-a\tan^2\beta). \]
The circle of curvature of the rectangular hyperbola \(x^2-y^2=a^2\) at the point \((a\operatorname{cosec}\theta, a\cot\theta)\) meets the curve again at \((a\operatorname{cosec}\phi, a\cot\phi)\). Prove that \[ \tan^4\frac{\theta}{2} \tan\frac{\phi}{2} = 1. \]
\(V\) is a given point on a given conic. Any chords \(VP, VQ\) are drawn, equally inclined to a given straight line. Prove that \(PQ\) passes through a fixed point.
If two conics have each double contact with a third, prove that their chords of contact with the third conic, and a pair of their chords of intersection with each other will all meet in a point and form a harmonic pencil.
Prove that if the equations \[ cy^2-2fyz+bz^2=0, \quad az^2-2gzx+cx^2=0, \quad bx^2-2hxy+ay^2=0 \] are satisfied by values of \(x,y,z\) all different from zero, then \[ abc+2fgh-af^2-bg^2-ch^2=0. \] Shew that when this condition is satisfied the equations are, in general, satisfied by two sets of values of \(x,y,z\) not proportional to one another.