Topic 2 [312 marks] 1 The rectangle ABCD is inscribed in a circle Sides [AD] and [AB] have lengths [12 marks] 3 cm and (\9\) cm respectively E is a point on side [AB] such that AE is 3 cm Side [DE] is produced to meet the circumcircle of ABCD at point P Use Ptolemy s theorem to calculate the length of chord [AP] 2 ABCD is a quadrilateral (AD) and (BC) intersect at F and (AB) and (CD) intersect at H (DB) and (CA) intersect (FH) at G and E respectively This is shown in the diagram below Prove that = HG GF HE EF The figure shows a circle C 1 with centre O and diameter [PQ] and a circle C 2 which intersects (PQ) at the points R and S T is one point of intersection of the two circles and (OT) is a tangent to C 2 3a Show that OR OT = OT OS [2 marks] Show that 3b PR RQ = 2OR PR RQ PR+RQ Show that = PS SQ PS+SQ
The points D, E, F lie on the sides [BC], [CA], [AB] of the triangle ABC and [AD], [BE], [CF] intersect at the point G You are given that CD = 2BD and AG = 2GD 4a By considering (BE) as a transversal to the triangle ACD, show that CE EA = 3 2 [2 marks] 4b Determine the ratios ; AF FB BG GE 4c The diagram shows a hexagon ABCDEF inscribed in a circle All the sides of the hexagon are equal in length The point P lies on the minor arc AB of the circle Using Ptolemy s theorem, show that PE + PD = PA + PB + PC + PF Triangle ABC has points D, E and F on sides [BC], [CA] and [AB] respectively; [AD], [BE] and [CF] intersect at the point P If 3BD = 2DC and CE = 4EA, calculate the ratios 5a AF : FB [4 marks] AP : PD 5b [4 marks] 6a Prove that the interior bisectors of two of the angles of a non-isosceles triangle and the exterior bisector of the third angle, meet the sides of the triangle in three collinear points An equilateral triangle QRT is inscribed in a circle If S is any point on the arc QR of the circle, 6b prove that ST = SQ + SR ; show that triangle RST is similar to triangle PSQ where P is the intersection of [TS] and [QR]; (iii) using your results from parts and deduce that 1 = + 1 SP SQ 1 SR [10 marks] PL PF = PM PG
A triangle 7a T has sides of length 3, 4 and 5 Find the radius of the circumscribed circle of T Find the radius of the inscribed circle of T A triangle 7b U has sides of length 4, 5 and 7 Show that the orthocentre, H, of U lies outside the triangle Show that the foot of the perpendicular from H to the longest side divides it in the ratio 29 : 20 8a Show that the opposite angles of a cyclic quadrilateral add up to 180 [3 marks] 8b A quadrilateral ABCD is inscribed in a circle S The four tangents to S at the vertices A, B, C and D form the edges of a quadrilateral EFGH Given that EFGH is cyclic, show that AC and BD intersect at right angles The circle C has centre O The point Q is fixed in the plane of the circle and outside the circle The point P is constrained to move on the circle 8c Show that the locus of a point P, which satisfies QP = kqp, is a circle C, where k is a constant and 0 < k < 1 Show that the two tangents to 8d C from Q are also tangents to C [4 marks] 9 The triangle ABC is isosceles and AB = BC = 5 D is the midpoint of AC and BD = 4 Find the lengths of the tangents from A, B and D to the circle inscribed in the triangle ABD
10a The diagram shows the line l meeting the sides of the triangle ABC at the points D, E and F The perpendiculars to l from A, B and C meet l at G, H and I State why = AF FB (iii) AG HB Hence prove Menelaus theorem for the triangle ABC State and prove the converse of Menelaus theorem [13 marks] 10b A straight line meets the sides (PQ), (QR), (RS), (SP) of a quadrilateral PQRS at the points U, V, W, X respectively Use Menelaus theorem to show that PU QV RW SX = 1 UQ VR WS XP 11 The diagram below shows a quadrilateral ABCD and a straight line which intersects (AB), (BC), (CD), (DA) at the points P, Q, R, S respectively Using Menelaus theorem, show that BQ CR DS = 1 AP PB QC RD SA The area of an equilateral triangle is 1 cm 2 Determine the area of: the circumscribed circle 12a the inscribed circle 12b [3 marks]
The points A, B have coordinates ( 1, 0), ( 0, 1) respectively The point P( x, y) moves in such a way that AP = kbp where k R + When 12c k = 1, show that the locus of P is a straight line [3 marks] 12d When k 1, the locus of P is a circle Find, in terms of k, the coordinates of C, the centre of this circle Find the equation of the locus of C as k varies [9 marks] 13 The parabola P has equation y 2 U (a V (a = 4ax The distinct points u 2 v 2, 2au) and, 2av) lie on P, where u, v 0 Given that UOV is a right angle, where O denotes the origin, (a) show that v = 4 ; μ [9 marks] (b) find expressions for the coordinates of W, the midpoint of [UV], in terms of a and u; (c) show that the locus of W, as u varies, is the parabola P with equation y 2 = 2ax 8a 2 ; (d) determine the coordinates of the vertex of P 14 ABCDEF is a hexagon A circle lies inside the hexagon and touches each of the six sides Show that AB + CD + EF = BC + DE + FA [5 marks]
15 (a) The function g is defined by g(x, y) = x 2 + y 2 + dx + ey + f C 1 has equation g(x,y) = 0 C 1 d 2 Show that the centre of has coordinates (, ) and the radius of C 1 is + f The point P(a, b) lies outside C 1 P to and the circle Show that the length of the tangents from C 1 is equal to g(a, b) (b) The circle C 2 has equation x 2 + y 2 6x 2y + 6 = 0 The line y = mx meets C 2 R and S R and S d 2 4 at the points Determine the quadratic equation whose roots are the x-coordinates of Hence, given that L denotes the length of the tangents from the origin O to C 2 e 2 e 2 4, show that OR OS = L 2 [12 marks]
The diagram above shows the points P(x, y) and P ( x, y ) which are equidistant from the origin O The line (OP) is inclined at an angle α to the x-axis and P OP = θ 16 (a) By first noting that OP = xsecα, show that x y = xcosθ y sin θ and find a similar expression for Hence write down the 2 2 matrix which represents the anticlockwise rotation about O which takes P to P' (b) The ellipse E has equation 5 x 2 + 5y 2 6xy = 8 Show that if E is rotated clockwise about the origin through 45, its equation becomes + y 2 = 1 x 2 4 E Hence determine the coordinates of the foci of [14 marks]
O ABC (AO), (BO), (CO) (BC), (CA), (AB) D, E, F (EF), (BC) G The diagram above shows a point inside a triangle The lines meet the lines at the points respectively The lines meet at the point (a) Show that, with the usual convention for the signs of lengths in a triangle, 17 BD BG = DC GC (b) The lines (FD), (CA) meet at the point H and the lines (DE), (AB) meet at the point I Show that the points G, H, I are collinear [14 marks] The point P(x, y) moves in such a way that its distance from the point ( 1, 0) is three times its distance from the point ( 1, 0) Find the equation of the locus of P 18a [4 marks] Show that this equation represents a circle and state its radius and the coordinates of its centre 18b [4 marks] The diagram shows triangle ABC together with its inscribed circle Show that [AD], [BE] and [CF] are concurrent 19a
PQRS is a parallelogram and T is a point inside the parallelogram such that the sum of 19b PTQ and RTS is 180 Show that TP TR + ST TQ = PQ QR [13 marks] The point 20a T(a t 2,2at) lies on the parabola y 2 = 4ax Show that the tangent to the parabola at T has equation x y = + at t [3 marks] The distinct points 20b P(a p 2,2ap) and Q(a q 2,2aq), where p, q 0, also lie on the parabola y 2 = 4ax Given that the line (PQ) passes through the focus, show that pq = 1 ; the tangents to the parabola at P and Q, intersect on the directrix 21a Prove the internal angle bisector theorem, namely that the internal bisector of an angle of a triangle divides the side opposite the angle into segments proportional to the sides adjacent to the angle 21b The bisector of the exterior angle A of the triangle ABC meets (BC) at P The bisector of the interior angle B meets [AC] at Q Given that (PQ) meets [AB] at R, use Menelaus theorem to prove that (CR) bisects the angle ACB In the acute angled triangle ABC, the points E, F lie on [AC], [AB] respectively such that [BE] is perpendicular to [AC] and [CF] is perpendicular to [AB] The lines (BE) and (CF) meet at H The line (BE) meets the circumcircle of the triangle ABC at P This is shown in the following diagram 22a Show that CEFB is a cyclic quadrilateral Show that HE = EP
22b The line (AH) meets [BC] at D By considering cyclic quadrilaterals show that CAD = EFH = EBC Hence show that [AD] is perpendicular to [BC] 23a Given that the elements of a 2 2 symmetric matrix are real, show that the eigenvalues are real; the eigenvectors are orthogonal if the eigenvalues are distinct [11 marks] 23b The matrix A is given by Find the eigenvalues and eigenvectors of A 11 3 A = ( ) 3 9 23c The ellipse E has equation X T AX = 24 where X = ( x ) and y A is as defined in part (b) Show that E can be rotated about the origin onto the ellipse E having equation 2 x 2 + 3 y 2 = 6 Find the acute angle through which E has to be rotated to coincide with E International Baccalaureate Organization 2017 International Baccalaureate - Baccalauréat International - Bachillerato Internacional