The tangents to a circle at A and B meet in T, and any line drawn through T cuts the circle in C and D. Prove that AC:AD = BC:BD, and that the tangents at C and D meet on AB.
The normal to an ellipse at a point P cuts the major axis in G. Prove that PG varies as the length of the diameter parallel to the tangent at P. Prove that, if the tangents to an ellipse at P and Q meet in T, and if the normals at P and Q meet the major axis in G and K, then the angle GTK has the same bisectors as the angle PTQ.
Prove that, if \(\alpha, \beta\) are the roots of the equation \(x^2-2px+q=0\), where \(p^2>q\), the condition that all the roots of the equation \[ x^3 - 3px^2 + 3qx - r = 0 \] are real, is that \(r\) must lie between \[ pq-2\alpha(p^2-q) \text{ and } pq-2\beta(p^2-q). \]
Prove that \[ 1 - \frac{3^3}{1} + \frac{5^3}{1\cdot 2} - \frac{7^3}{1\cdot 2\cdot 3} + \dots = \frac{3}{e}. \]
Prove that, if \(\alpha+\beta+\gamma=2\sigma\), \begin{align*} \sin 3(\sigma-\alpha)\sin(\beta-\gamma) + \sin 3(\sigma-\beta)\sin(\gamma-\alpha) + \sin 3(\sigma-\gamma)\sin(\alpha-\beta) \\ = 4\sin\sigma\sin(\beta-\gamma)\sin(\gamma-\alpha)\sin(\alpha-\beta). \end{align*}
Points P, Q, R are taken in the sides AB, BC, CA (respectively) of a triangle ABC such that the triangle PQR is similar to the triangle ABC. Prove that if \(\theta\) denotes the angle CQR, \[ QR:BC = \sin\omega : \sin(\theta+\omega), \] where \[ \cot\omega = \cot A + \cot B + \cot C. \] Determine the value of \(\theta\) when the area of the triangle PQR is a minimum.
Through a point P two lines are drawn in given directions. Prove that, if the line joining the middle points of the intercepts made by these lines on two fixed perpendicular lines OA, OB is perpendicular to the line OP, the locus of P is either of two fixed perpendicular lines.
The tangents to the parabola \(y^2=4ax\) at P and Q meet in T. If \((\alpha, \beta)\) are the coordinates of T, find the coordinates of the orthocentre of the triangle TPQ. A variable parabola has a fixed vertex O and a fixed axis; and from a fixed point T tangents TP, TQ are drawn to the curve. Prove that the locus of the orthocentre of the triangle TPQ is a rectangular hyperbola which passes through O and T.
Prove that the points of contact of the tangents drawn to the conic \(b^2x^2+a^2y^2=a^2b^2\) from two points \((\alpha,\beta)(\alpha',\beta')\) lie on the conic \[ \left(\frac{\alpha x}{a^2}+\frac{\beta y}{b^2}-1\right)\left(\frac{\alpha'x}{a^2}+\frac{\beta'y}{b^2}-1\right) = \left(\frac{\alpha\alpha'}{a^2}+\frac{\beta\beta'}{b^2}-1\right)\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1\right). \] Hence, or otherwise, prove that, if from a fixed point T tangents TP, TQ are drawn to any conic of a given confocal system, the circle TPQ passes through a second fixed point.
The diameters of the top and bottom sections of a conical bucket are 12 inches and 6 inches. The bucket is filled with water which leaks out through a small hole in the bottom at the rate which varies as the square root of the depth of the water. In 10 seconds the diameter of the free surface of the water is reduced by 2 inches; shew that the rest of the water will leak away in 49 seconds approximately.