By Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

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Obtain the equations of the circles which touch the *-axis at (5, 0) and make an intercept of 24 on the >>-axis. 32. Condition that a line be tangent to a circle Tf the straight line lx+my+n = 0 is a tangent to the circle x2+y*+2gx+2fy+c = 0 the perpendicular from the centre (—g, —f) to the line lx+my+n = 0 is equal to the radius \/{g2+f2—c). +ft ) and so (-Ig-mf+nf = (/ 2 +m 2 ) ( g 2 + / 2 - c ) which reduces to (c-f2)l2+2fglm+(c-g2)m2-2fmn-2gnl+n2 = 0. This formula is too cumbersome to be remembered.

X2—y2) — 2(lxm2—i2m^) (l1l2—fn1m2)xy = 0. Since the lines of the line-pair are distinct, l1m2—l2m1 Φ 0 and so division by lxm2-l2m1 yields {iim2+l2m^) (x2—y2) — 2(l1I2—m1m2)xy = 0. Substituting for lj2, m1m2 and Ιχηί2-{-12ηι^ we have h(x2-y2) - (a-b)xy = 0. EXAMPLES 18. Write down the equation of the line-pair bisecting the angles between the line-pairs (i) x2—y2 = 0; (ii) 4x2—xy—3y2 = 0; (iii) x2 cos Ö+2^-^ 2 sin Θ = 0. 19. Show that x—y = 0 bisects the angle between the lines Ax2— \\xy+4y2 = 0 and write down the equation of the other bisector.

Cf and so αχχ+by^+c d=± The sign of d is indeterminate, but from the previous section we see that the lengths of the perpendiculars from points on the same side of the straight line have the same sign. FIG. 14 If the equation is written in the normal form x cos α+j; sin a=p, the perpendicular distance from (xl9 νλ) is ± (*i COS a + J i Sin a—p). EXAMPLES 49. Calculate the perpendicular distances from (2, —1) to the straight lines; (i) 4x+3y = 2; (ii) * - ^ + 1 = 0; (iii) 2x-3y-l = 0. STRAIGHT LINE 29 50.