Calendar: 1871-1872 Page 582

582 THE SCHOOL Show that the centre of the inscribed circle is the ortho centre of the triangle formed by joining the centres of the escribed circles 11 Prove Demoivre's theorem and show that if מ be an integer β cos -Ι- ν sin will have η different values and not more η when cos θ and sin θ are given 12 If 2R be the diameter of the circle circumscribing given triangle prove that sin Λ sin Β sin If R2 R3 be the radii of the three circles described about the quadrilateral figures formed by perpendiculars from the angular points upon the opposite sides then Λ22 R32 R2 2RlRi Ry 13 Obtain the exponential values of sin θ and cos Θ Find the real part of the expression sin β 14 Prove the formula ע ק- פ-י and show that the sum of the squares of the reciprocals of the natural numbers is tonutrtcal Conic Sections If the tangent at any point Ρ of conic section meet the directrix in prove that SZ is at right angles to SP The semi latus rectum is an harmonic mean between the two segments of any focal chord Draw pair of tangents to parabola from an external point Describe parabola which has given focus and two given tangonts