10273 problems found
A straight line \(MN\) of given length slides with its ends \(M,N\) on two fixed straight lines \(OX, OY\). \(PM, PN\) are perpendicular to \(OX,OY\) respectively. Find the locus of \(P\).
\(Q\) is any point on the polar of \(P\) with respect to a given circle. Prove that the circle on \(PQ\) as diameter cuts the given circle orthogonally.
Prove that the inverse of a circle with respect to a point not in its plane is a circle. Two circles in different planes both touch the line of intersection of the planes at the same point. Shew how to find the point \(O\) with respect to which the circles may be inverted into each other, and prove that every plane that touches both circles passes through \(O\).
\(TP, TQ\) are tangents at \(P\) and \(Q\) to a parabola whose focus is \(S\). Prove that the angles \(PTQ\) and \(PST\) are supplementary. Three tangents to a parabola form an equilateral triangle. Prove that the focal radii to the vertices of the triangle pass each through the point of contact of the opposite side.
The normal at \(P\) to an ellipse cuts the major axis in \(G\), and \(CF\) is the perpendicular from the centre \(C\) on the normal. Prove that \(PF.PG\) is equal to the square on the semi-minor axis. Prove also that if \(S\) and \(S'\) are the foci, the circle \(SPS'\) cuts the tangent and normal at \(P\) in points which lie on the minor axis, produced if necessary.
Find the condition that two circles whose equations are given should cut each other at right angles. If the origin is one of the limiting points of a system of coaxal circles of which \[ x^2+y^2+2gx+2fy+c=0 \] is a member, prove that the equation of the system of circles that cuts them all at right angles is \[ (x^2+y^2)(g+\mu f)+c(x+\mu y)=0. \]
Find the equation of the normal to the parabola \[ y^2=4ax \] at the point \(P(am^2, 2am)\). From a variable point on the fixed normal at \(P\) two other normals \(QR, QS\) are drawn to the parabola. Prove that the line \(RS\) joining their feet has a fixed direction.
\(OP, OQ\) are any two conjugate diameters of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2}=1\), and meet the fixed line \(lx+my=1\) in \(P\) and \(Q\). \(PR, QR\) are perpendicular to \(OP,OQ\) respectively. Prove that the locus of \(R\) is \[ a^2lx+b^2my=a^2+b^2. \]
Show that the feet of the perpendiculars from a point \(P\) on the circumcircle of a triangle lie on a straight line (\(\lambda\)). Prove that if \(\lambda_1, \lambda_2, \lambda_3\) correspond to \(P_1, P_2, P_3\) the triangle \(P_1P_2P_3\) is similar to that formed by \(\lambda_1, \lambda_2, \lambda_3\).
Prove that the radical axis of a fixed circle and a circle which passes through two given points passes through a fixed point. Reciprocate this theorem with respect to a circle.