Plane and solid analytic geometry; an elementary textbook . *-* b -bxa Fig. 93. Substituting these val-ues in (3), it reduces to a? ,y^_ 1a2 b2~2 10. Find the locus of the vertex of a triangle whosebase is a line joining the foci and whose sides are parallelto two conjugate diameters. 11. Find the locus of the middle point of chordsdrawn through a fixed point in the (a) ellipse, (£>)parabola. 12. Tangents are drawn to the parabola y2= the locus of their pole with respect to the circlex2 + y2 = r2. 13. The two circles x2 + y2 = a2 and x2 + y2 — ax = 0are tangent internally. Find the
Plane and solid analytic geometry; an elementary textbook . *-* b -bxa Fig. 93. Substituting these val-ues in (3), it reduces to a? ,y^_ 1a2 b2~2 10. Find the locus of the vertex of a triangle whosebase is a line joining the foci and whose sides are parallelto two conjugate diameters. 11. Find the locus of the middle point of chordsdrawn through a fixed point in the (a) ellipse, (£>)parabola. 12. Tangents are drawn to the parabola y2= the locus of their pole with respect to the circlex2 + y2 = r2. 13. The two circles x2 + y2 = a2 and x2 + y2 — ax = 0are tangent internally. Find the locus of the centresof circles which are tangent to both the given circles. Ch. XIV] PROBLEMS IN LOCI 189 Let the two circles be drawn, and let (V, y) be thecentre of any circle which is tangent to both the lines OP mustpass through B, the pointof contact of the twocircles, and OP mustpass through 0. Hence, P0=00-OP= r- OP,and PB = P0-BO= P0 --> But P C and PB are radii of the same circle. Hence, r-OP =P0. Fig. 94. Or f-v. x2 4- y2 4 + y 2 Squaring and reducing, the equation of the locus re-duces to 8 x2 + 9 y2 — 4 rx — 4 r2 = 0. What curve is thisand how is it situated? 14. Find the locus of the centres of all circles whichpass through the point (0, 3) and are tangent internallyto x2 + y2 = 25. 15. Find the locus of the centres of circles which aretangent to a given circle and pass through a fixed pointoutside of that circle. 16. Lines are drawn from the point (1, 1) to thehyperbola x2 — y2 = 1. Find the locus of the points whichdivide these lines in the ratio of 2 to 1. 190 ANALYTIC GEOMETRY [Ch. XIV 17. Lines are drawn from the centre 0 of the circlex2 + y2 = r2, cutting the circle in A and the line, x = a,in B. Find the locus of P, if 0, A, B, and P form aharmonic range. Show that the result will represent anellipse, hyperbola, or parabola, according as 4 r2 a2, 4 r2 = a2. 18. Find the locus of the vertex of a triangle if thelength of the base is c,
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