By Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)
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Extra resources for Analytical Geometry
This equation is homogeneous of the second degree and so repre sents a line-pair through the origin. To obtain the points of inter section of this line-pair with u = 0, substitute Ix+my = —n and 44 ANALYTICAL GEOMETRY the equation then reduces to S = 0. Thus the line-pair is cut by u = 0 at the points given by S = u = 0. Hence the equation represents the pair of lines joining the origin to the points of intersection of S = 0 and u = 0. Note carefully that this result is valid even if S = 0 does not represent a line-pair.
15 That is, lx—y+6 = ± 5(x+y+2) and so we obtain the equations x—3y—2 = 0 and 3x+y+4 — 0. The first equation gives the bisector of the obtuse angle. EXAMPLES 54. Obtain the equation of the bisector of the acute angle between the pair of lines: (i) x+2y = 1, 2x +y+3=0; (ii) 3x-4y = 5, -5*+12j> = 2. 55. Obtain the coordinates of the centre of the circle inscribed in the triangle whose vertices are at (—7, —5), (17, 1) and (1,14). Further, calculate the radius of this inscribed circle. 56. Show that the point (3, — 1) is equidistant from the lines 3ΛΓ—Ay—16=0 and 4χ+3>>-12 = 0.
7. Find the coordinates of the circumcentre of the triangle formed by the straight lines 3*-j>-5 = 0, x+2j>-4 = 0 and 5jt+3j> + l = 0. 36 ANALYTICAL GEOMETRY 8. Find the equation of the locus of a point equidistant from (xu y±) and 9. Show that the line joining the points (xl9 yd and (x2, y2) will subtend a right angle at (x3,yz) if (*s—*i) (*s—*2> + (ys—yù (y*—yd = 0. Hence, obtain the equation of the circle on the line joining (xl9 yd and (x2, y2) as diameter. 10. Prove that the straight line (A+2)*+(3A-l)j;+A = 0, where λ is a variable, passes through a fixed point and find its coordinates.
Analytical Geometry by Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)