...chords of a circle, which subtend a right angle at a fixed point ; and thence show that the envelope of a system of chords of a conic, which subtend a right angle at the focus, is another conic of coincident focus ; and that the locus of the middle points of a system...

...the chord of contact of which subtends a right angle at a fixed point j and also of the middle'points of a system of chords of a circle, which subtend a...fixed point is another conic. Solution by the Rev. J. WOLSTJJNIIOLMK, MA Taking the fixed point for origin, let equation of circle be (x— c)" + j/2 = a2,...

...and hence, or in any other way, determine the directions of the axes of the conic. 14. Prove that the chords of a conic which subtend a right angle at a fixed point P of the conic all pass through a certain fixed point of the normal at the point P. In the case of...

...of the parabola which subtend a right angle at S all touch a conic whose centre is C. 38. Shew that chords of a conic which subtend a right angle at a fixed point 0 envelope another conic. Shew also that the point 0 is a focus of the envelope and that the directrix...

...corresponding normal as axes of z and y respectively, is ax2 + 2hxy + by2 + 2/y = 0. Ex. 1. All cliords of a conic which subtend a right angle at a fixed point 0 on the conic, cut the normal at O in a fixed point. Take the tangent and normal at 0 for axes ; then...

...= 0 are(0, -}) and (-f -|). 18. The focus of the parabola is the point (— -J, — •§). 19. All chords of a conic which subtend a right angle at a fixed point on a conic, cut the normal at 0 in a fixed point. Solution. Take the tangent and normal at 0 for the...

...generally of a conic at any point, by changing the origin to the point and back again. (3) Prove that chords of a conic, which subtend a right angle at a fixed point O in the conic, pass through a fixed point on the normal at O. 152. Asymptotes. To determine the nature...

...touch a fixed conic, whose foci are the fixed point and the centre of the circle. 15. The envelope of chords of a conic which subtend a right angle at a fixed point О is a conic, whose focus is О and directrix the polar of O. 16. If a straight line is cut harmonically...

...0 meets the hyperbola again in C', the tangents to the hyperbola at C, C' intersect on AB. 19. Find the locus of the middle points of a system of chords of a hyperbola passing through a fixed point on one of the asymptotes. 20. OP, CD are conjugate semi-diameters;...

...parallel chords be drawn in a circle, their middle points lie in the same straight line. 61. What is the locus of the middle points of a system of chords of given length in a circle ? 62. If two circles are tangent to each other, any straight line drawn through...