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

ISBN-10: 0080099734

ISBN-13: 9780080099736

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24. Show that ax+by+c = 0 is a tangent to the circle x2~\-y2 = r2 if 2 2 r (a +b2) = c2. 25. Find the condition that y = mx+c intersects the circle x2~\-y2 = r2 in two distinct real points. 30. JoachimsthaPs equation Let the two points Ax(xl9 yj and A2(x2, y2) intersect the circle S ΞΞ x*+y*+2gx+2fy+c = 0 at Λ and P2 (Fig. 19). The coordinates of the point P which divides AXA2 in the ratio λ2/λ1 (section 4) are ( ( λ Λ + λ ^ / ί λ ^ λ2), ( λ Λ + ^)Κλχ+ λ2) ). 53 CIRCLE If this point P lies on the circle, we have \ λ 1+ λ 2 / \ λ 1+ λ 2 / \ λ χ +λ 2 y ^ λ 1 +λ 2 / + On multiplication by (Λ+λ,,)2, this equation simplifies to FIG.

A further calculation yields (l1m2-l2m1)2=4(h2-ab) ; K « 2 - ^ ) 2 = 4(f2-bc (/i"2-4"i) 2 = 4(g 2 -™), ; and so it is also necessary that h2 > ab ; f2 > be ; g2 > ca. It can be shown that these inequalities are not independent when Δ=0. The complete result (which we shall not attempt to prove) is that the necessary and sufficient condition that S = 0 represent a line-pair is that Δ = 0, h2 > ab or Δ = 0, h2 = ab, f2+g2 > c{a+b). Illustration: If S = ax2+2hxy+by2+2gx+2fy+c = 0 represents a pair of non-parallel straight lines, show that the coordinates of the point of intersection of the line-pair satisfy the three equations M] ΞΞ ax+hy+g = 0, ih=hx+by+f=09 «3 = gx+fy+c = 0.

If a = 0, the equation 2hxy+by2 = 0 represents the two straight lines y = 0 and 2hx+by = 0. EXAMPLES 1. Find the equations of the lines represented by the following equations: (i) Ax2-y2 = 0; (ii) 2x2-5xy-3y2 = 0; (iii) 4x2-20xy+25y2 = 0; (iv) 3χ2+4χ>> = 0. 2. Form the equations which represent the following pairs of lines: (i) y = 0, Ay = x; (ii) 3x-y = 0, x+3y = 0; (iii) x = 0, y = 0; (iv) y = nx, y = mx. 3. Find the angle between the straight lines represented by mxm^x2— (m1-\-m2)xy-{-y2 = 0 and hence prove that the angle between the lines given by ax2+2hxy+by2 = 0 is tan" 1 2^(h 2 -ab)/(a+b).

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


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