If the polars of two points P, Q meet in R, then R is the pole of the line PQ. For, since R is on the polar of P, the polar of R will go through P; similarly the polar of R will go through Q; and therefore it must be the line PQ. If any chord of a conic be drawn through a fixed point Q, and P be the pole of the chord; then, since is on the polar of P, the point P will always lie on a fixed straight line, namely on the polar of Q. Def. Two points are said to be conjugate with respect to a conic when each lies on the polar of the other. Def. Two straight lines are said to be conjugate with respect to a conic when each passes through the pole of the other. Conjugate diameters, as defined in Art. 127, are conjugate lines through the centre. 181. If any chord of a conic be drawn through a point O it will be cut harmonically by the curve and the polar of 0. Let OPQR be any chord which cuts a conic in P, R and the polar of O with respect to the conic in Q. Take O for origin, and the line OPQR for axis of x; and let the equation of the conic be Where ax2+2hxy + by2 + 2gx+2fy + c = 0. y=0 cuts the conic we have 182. To find the locus of the middle points of a system of parallel chords of a conic. Let (x, y) and (z”, y'') be two points on the conic. The equation a (x − x') (x − x'') + h {{(x − x) (y− y') + (x − x′) (y—y')} +b (y− y) (y—y')=ax2+2hxy+by+2gx+2ƒy +c ...(i) is the equation of the straight line joining the two points. In (i) the coefficient of x is a (x+x") + h (y′+y′′)+2g, and the coefficient of y is h(x′+x′′) + b (y' + y'') + 2ƒ ; hence if the line is parallel to the line y = mx, we have a (x′+x′′)+h{y' + y′′) + 2g ́h (x+x") + by + y) + 2ƒ m ..(ii). Now, if (x, y) be the middle point of the chord joining (x', y') and (x", y′), then 2x=x' +x', and 2y = y' + y′′; therefore, from (ii), we have or x (a + mh) + y (h + mb) +g+mƒ= 0.......(iii), which is the required equation. If the line (iii) be written in the form y=mx+k, then we have or a + h (m + m') + bmm' = 0 .......................(iv). This is the condition that the lines y = mx and y = m'x may be parallel to conjugate diameters of the conic given by the general equation of the second degree. (w). 183. To find the condition that the two lines given by the equation Ax2 +2Hxy + By3 =0 may be conjugate diameters of the conic ax2+2hxy + by2 = 1. If the lines given by the equation Ax2+2Hxy+By2= 0 be the same as y — mx = 0, and y — m'x=0; then if But y-mx=0 and y-m'x=0 are conjugate diameters a + h (m + m') + bmm' = 0. Therefore the required condition is [The above result follows at once from Articles 155 and 58.] Ex. 1. To find the equation of the equi-conjugate diameters of the conic ax2+2hxy+by2 = 1. The straight lines through the centre of a conic and any concentric circle give equal diameters. Through the intersections of the conic and the circle whose equation is λ (x2 + y2+ 2xy cos w) = 1, the lines (a−λ) x2 + 2 (h−λ cos w) xy + (b −λ) y2=0 pass. These are conjugate if b (a−λ) + a (b −λ)=2h (h−λ cos w). Substituting the value of λ so found, we have the required equation ax2+2hxy+by2 – 2 (ab-h2) (x2+ y2+2xy cos w) = 0. Ex. 2. To shew that any two concentric conics have in general one and only one pair of common conjugate diameters. Let the equations of the two conics be ax2+2hxy+by2=1, and a'x2+2h'xy+b'y2=1. The diameters Ax2+2Hxy+By2=0 are conjugate with respect to both conics if The equation of the common conjugate diameters is therefore (ha'ah') x2 - (ab' — a'b) xy + (bh' — b'h) y2 = 0. Since any two concentric conics have one pair of conjugate diameters in common, it follows that the equations of any two concentric conics can be reduced to the forms ax2+by2=1, a'x2+b'y2=1. 184. To find the length of a straight line drawn from a given point in a given direction to meet a conic. Let (x, y) be the given point, and let a line be drawn through it making an angle with the axis of x. The point which is at a distance r along the line from (x, y) is (x+rcos 0, y' +rsin ), the axes being supposed to be rectangular; and, if this point be on the conic given by the general equation, we have a (x'+r cos 0)2+2h (x' + r cos 0) (y′+ r sin 0)+b(y'+rsinė)": +2g (x+r cos 0)+2f(y+r sin ) + c = 0, or 2 (a cos 0+2h sin cos 0+b sin2 0) +2r cos(ax+hy+g) + 2r sin 0 (hx'+by'+f)+$(x', y')=0. The roots of this quadratic equation are the two values of r required. 185. If the point (x, y) be the middle point of the chord intercepted by the conic on the line, the two values of r, given by the quadratic equation in the preceding Article, will be equal in magnitude and opposite in sign; hence the coefficient of r must vanish; thus (ax+hy+g) cos 0+ (hx' + by +ƒ) sin 0 = 0. If the chords are always drawn in a fixed direction, so that is constant, the above equation gives us the relation satisfied by the co-ordinates x, y' of the middle point of any chord. The locus of the middle points of chords of the conic which make an angle @ with the axis of x is therefore a straight line. [See Art. 182.] 186. The rectangle of the segments of the chord which passes through the point (x, y) and makes an angle with the axis of x, is the product of the two values of r given by the quadratic equation in Art. 184; and is equal to $ (x', y') a cos20+2h sin cos 0+b sin20 Cor. 1. If through the same point (x, y) another chord be drawn making an angle with the axis of x, the rectangle of the segments of this chord will be $ (x', y') a cos2 '+2h sin e' cos 0' + b sin2 0' * seg Hence we see that the ratio of the rectangles of the ments of two chords of a conic drawn in given directions through the same point is constant for all points, including the centre of the conic, so that the ratio is equal to the ratio of the squares of the parallel diameters of the conic. Cor. 2. The ratio of the two tangents drawn to a conic from any point is equal to the ratio of the parallel diameters of the conic. Cor. 3. If through the point (x", y') a chord be drawn also making an angle with the axis of x, the rectangle of the segments of this chord will be (x", y") a cos2 0 + 2h sin 0 cos 0 + b sin2 0 * Hence the ratio of the rectangles of the segments of any two parallel chords drawn through two fixed points (x', y') and (x", y') is constant and equal to the ratio of (x', y') to p(x”, y'). Cor. 4. If a circle cut a conic in four points P, Q, R, S, the line PQ joining any two of the points and the line RS joining the other two make equal angles with an axis of the conic. For, if PQ and RS meet in T, the rectangles TP. TQ and TR. TS are equal since the four points are on a circle. Therefore by Cor. 1, the parallel diameters of the conic are equal; and hence they must be equally inclined to an axis of the conic. Ex. 1. If a, ß, y, d be the eccentric angles of the four points of intersection of a circle and an ellipse, then will a+ß+y+d=2nπ. The equations of the lines joining a, ẞ and y, d are |