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Geometry Topology

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

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14) from the point /'(xiJi) to the straight line ax+by+c = 0. The equation of the straight line PN is b(x-xj) - a{y-yù = 0. Let the coordinates of N be (α, β). Then A(a-*i) - a(ß-yu Since N lies on ax+by+c can be written = 0. = 0, we have aa+bß+c <*(<>>—xd+Wß—yu = = 0 which —tßXi+byi+c). +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).

51. Find the radius of the circle with centre at (— 1, —3) and which touches the line 3x+2j>-4 = 0. 52. Write down the perpendicular distances of the point (2, 1) from the parallel lines 3x—4y+4 = 0 and Ay—3x+5 = 0 and hence determine the distance between these lines. 53. Obtain the coordinates of the centroid of the triangle with sides along the lines x+y— 1 = 0, x—y— 1 = 0 and *—3j>-f3 = 0. 9) lies inside the triangle. ) 18. Bisectors of angles between two straight lines The perpendicular distances from any point on the internal or external bisector of the angle between two straight lines to the lines themselves are equal numerically.

In this case show that the locus of the foot of the perpendicular from the origin to the line lx+my+n=0 has the equation x2 +y2+2gx+2fy+c = 0. 7. Show that all chords of the locus ax2+2hxy-\-by2+2gx+2fy+c = 0 which subtend a right angle at the origin pass through a fixed point. 8. If the equation 2hxy+2gx+2fy+c =0 represents two straight lines, show that they form a rectangle with the axes, having the straight lines gx—fy = 0 and ghx+hfy+fg = 0 as diagonals. 9. Show that the product of the perpendicular distances from (α, β) to the (aa2+2haß+bß2)/\/{(a-b)2+4h2}.

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