10273 problems found
If \(S_r \equiv x^2+y^2+2g_rx+2f_ry+c_r\), interpret geometrically the following equations:
Define a parabola and deduce the parametric representation in the usual form \((at^2, 2at)\). \par A circle is drawn through the vertex of a parabola and cuts the curve in three other points \(P, Q, R\). Shew that the normals to the parabola at \(P, Q, R\) are concurrent. \par Shew also that if a circle is drawn through any two fixed points on a parabola, the midpoint of the join of the other pair of intersections lies on a fixed straight line.
Shew that of the family of confocal conics given by the equation \(\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1\), two and only two members pass through a given point, one a hyperbola and one an ellipse. Derive the expression for the coordinates \((x,y)\) of the point in terms of the parameters \(\lambda_1, \lambda_2\) corresponding to the two conics in the form: \[ x^2 = \frac{(a^2+\lambda_1)(a^2+\lambda_2)}{a^2-b^2}, \quad y^2 = \frac{(b^2+\lambda_1)(b^2+\lambda_2)}{b^2-a^2}. \] Shew that the semi-axis of the ellipse parallel to the normal at \((x,y)\) to the hyperbola is such that the square of its length is \(\lambda_1 \sim \lambda_2\).
Shew that \(x^2-2x\cos\theta+1\) is a factor of \(x^{2n}-2x^n\cos n\theta+1\), and find the other real quadratic factors of this expression. \par Hence, or otherwise, obtain the results: \begin{align*} \sin n\theta &= 2^{n-1} \prod_{s=0}^{n-1} \left[\sin\left(\theta+\frac{s\pi}{n}\right)\right]; \\ \cos n\theta &= 2^{n-1} \prod_{s=0}^{n-1} \left[\sin\left(\theta+\frac{(2s+1)\pi}{2n}\right)\right]. \end{align*}
Shew that a quadrilateral with sides of given lengths has its greatest area when it is cyclic. \par Shew further that the area of a cyclic quadrilateral is \(\sqrt{(s-a)(s-b)(s-c)(s-d)}\), where \(a,b,c,d\) are the lengths of the sides, and \(2s=a+b+c+d\).
Two variable points \(P, Q\) on a fixed line subtend a constant angle at a fixed point \(O\); prove that the variable circle \(OPQ\) touches a fixed circle, with respect to which \(O\) and the reflexion of \(O\) in the fixed line are inverse points.
``A tangent to a circle is perpendicular to the radius through its point of contact'': reciprocate this property with respect to any other circle. \par A variable tangent \(t\) to a conic meets a fixed tangent at \(P\); find the locus of intersection of \(t\) and the line through a focus \(S\) perpendicular to \(SP\).
(i) The two sets of points \(P_1, P_2, \dots\) on a line \(OX\), and \(Q_1, Q_2, \dots\) on a line \(OY\) are homographic, \(P_r\) and \(Q_r\) being corresponding points; prove that the intersections of pairs of lines such as \(P_rQ_s\) and \(P_sQ_r\) lie on a line. \par (ii) State the condition that the conic which is the envelope of \(P_rQ_r\) should be a pair of points. \(O, A, B\) are three fixed points on a fixed line, and \(O, P, Q\) are three fixed points on a line which rotates round \(O\) in a plane through the fixed line; prove that the locus of the intersection of \(AP, BQ\) is a circle and identify its centre and its radius.
Explain what is meant by the statement that two pairs of points on a conic are harmonic. \par \(O, X\) are two fixed points on a conic, and \(OP, OQ\) are variable chords of the conic equally inclined to \(OX\); prove that the chord \(PQ\) passes through a fixed point on the tangent at \(X\) to the conic.
Prove that the points of contact of the tangent lines from a point \(P\) to a sphere lie on a plane \(p\) (the polar plane of \(P\)), and that, if \(q\) is the polar plane of \(Q\), then the polar plane of any point on the line \(PQ\) is collinear with \(p, q\). \par Prove also that the lines \(PQ, pq\) are perpendicular to each other and that the feet of their common normal are inverse points with respect to the sphere. \item[] N.B. The equations in Questions 6, 7, 8, 9 are referred to rectangular Cartesian coordinate axes.