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geo7 - Problems in Geometry(7 1 Let be an ellipse with foci...

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Problems in Geometry (7) 1. Let ϕ be an ellipse with foci F, F 0 . If the tangent lines at A, B ϕ intersect in P , prove that PF AB iff F AB . 1 2. Let ϕ be an ellipse with foci F, F 0 . Let t, t 0 be the tangents to ϕ which intersect in P . If H, H 0 are respectively the feet of the perpendiculars from F on t, t 0 , prove that PF 0 is perpendicular to HH 0 . 2 3. Let ϕ be an ellipse of foci F, F 0 . Let ϕ FF 0 = { A, A 0 } . Let ‘, ‘ 0 be the tangents to ϕ. at A, A 0 . For any tangent t to ϕ at M ϕ let t = { P } , t 0 = { P 0 } . (A) Prove that < ( FP, FP 0 ) = < ( F 0 P, F 0 P 0 ) = π/ 2 3 . (B) Prove that FP, F 0 P 0 intersect on the normal to ϕ at M. 4 4. Let ϕ be an ellipse of foci F, F 0 . Consider X ϕ. Let Q, Q 0 be the points in which the normal of ϕ at X intersects the perpendiculars to XF, XF 0 erected at F, F 0 respectively. Prove that, the perpendicular bisector of [ FF 0 ] bisects [ Q, Q 0 ] . 5. Consider an ellipse ϕ with foci F, F 0 . Let a line through F meet ϕ in X, X 0 . If the normals to ϕ at X, X 0 intersect in N, prove that the parallel to FF 0 through N bisects [ XX 0 ] . 5 6. (A) Let
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