Edecel GCE A Level Maths Further Maths 3 Coordinate Sstems Edited b: K V Kumaran kumarmaths.weebl.com 1
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1. An ellipse has equation + 16 9 = 1. (a) Sketch the ellipse. (b) Find the value of the eccentricit e. (c) State the coordinates of the foci of the ellipse.. The hperbola C has equation 1. b a (a) Show that an equation of the normal to C at the point P (a sec t, b tan t) is (1) () () [P5 June 00 Qn 1] a sin t + b = (a + b ) tan t. (6) The normal to C at P cuts the -ais at the point A and S is a focus of C. Given that the eccentricit of C is 3, and that OA = 3OS, where O is the origin, (b) determine the possible values of t, for 0 t. (8) [P5 June 003 Qn 1] 3. An ellipse, with equation = 1, has foci S and S. 9 4 (a) Find the coordinates of the foci of the ellipse. (b) Using the focus-directri propert of the ellipse, show that, for an point P on the ellipse, 4. The hperbola H has equation 16 Find (a) the value of the eccentricit of H, (b) the distance between the foci of H. The ellipse E has equation 16 + 4 = 1. SP + S P = 6. = 1. [P5 June 004 Qn 3] 4 (c) Sketch H and E on the same diagram, showing the coordinates of the points where each curve crosses the aes. [FP/P5 Januar 006 Qn ] () () kumarmaths.weebl.com 6
5. The point S, which lies on the positive -ais, is a focus of the ellipse with equation Given that S is also the focus of a parabola P, with verte at the origin, find (a) a cartesian equation for P, (b) an equation for the directri of P. 4 + = 1. (1) 6. The ellipse E has equation a > 0. b + [FP June 006 Qn ] = 1 and the line L has equation = m + c, where m > 0 and c (a) Show that, if L and E have an points of intersection, the -coordinates of these points are the roots of the equation Hence, given that L is a tangent to E, (b + a m ) + a mc + a (c b ) = 0. () (b) show that c = b + a m. () The tangent L meets the negative -ais at the point A and the positive -ais at the point B, and O is the origin. (c) Find, in terms of m, a and b, the area of triangle OAB. (d) Prove that, as m varies, the minimum area of triangle OAB is ab. (e) Find, in terms of a, the -coordinate of the point of contact of L and E when the area of triangle OAB is a minimum. 7. The ellipse D has equation 5 + 9 = 1 and the ellipse E has equation [FP June 006 Qn 9] + = 1. 4 9 (a) Sketch D and E on the same diagram, showing the coordinates of the points where each curve crosses the aes. The point S is a focus of D and the point T is a focus of E. (b) Find the length of ST. [FP June 007 Qn ] kumarmaths.weebl.com 7
8. The hperbola H has equation 16 9 = 1. (a) Show that an equation for the normal to H at a point P (4 sec t, 3 tan t) is 4 sin t + 3 = 5 tan t. (6) The point S, which lies on the positive -ais, is a focus of H. Given that PS is parallel to the -ais and that the -coordinate of P is positive, (b) find the values of the coordinates of P. Given that the normal to H at this point P intersects the -ais at the point R, (c) find the area of triangle PRS. 8. The hperbola H has equation = 1, where a and b are constants. a b The line L has equation = m + c, where m and c are constants. [FP June 008 Qn 7] (a) Given that L and H meet, show that the -coordinates of the points of intersection are the roots of the equation Hence, given that L is a tangent to H, (a m b ) + a mc + a (c + b ) = 0. () (b) show that a m = b + c. The hperbola H has equation = 1. 5 16 () (c) Find the equations of the tangents to H which pass through the point (1, 4). 9. The line = 8 is a directri of the ellipse with equation (7) [FP3 June 009 Qn 6] a b and the point (, 0) is the corresponding focus. = 1, a > 0, b >0, Find the value of a and the value of b. [FP3 June 010 Qn 1] kumarmaths.weebl.com 8
10. The hperbola H has equation = 1. 16 4 The line l1 is the tangent to H at the point P (4 sec t, tan t). (a) Use calculus to show that an equation of l1 is sin t = 4 cos t The line l passes through the origin and is perpendicular to l1. The lines l1 and l intersect at the point Q. (b) Show that, as t varies, an equation of the locus of Q is 11. The hperbola H has equation ( + ) = 16 4 a b = 1. () [FP3 June 010 Qn 8] (a) Use calculus to show that the equation of the tangent to H at the point (a cosh θ, b sinh θ ) ma be written in the form b cosh θ a sinh θ = ab. The line l 1 is the tangent to H at the point (a cosh θ, b sinh θ ), θ 0. Given that l 1 meets the -ais at the point P, (b) find, in terms of a and θ, the coordinates of P. The line l is the tangent to H at the point (a, 0). () Given that l 1 and l meet at the point Q, (c) find, in terms of a, b and θ, the coordinates of Q. (d) Show that, as θ varies, the locus of the mid-point of PQ has equation (4 + b ) = ab. () (6) [FP3 June 011 Qn 8] kumarmaths.weebl.com 9
1. The hperbola H has equation Find (a) the coordinates of the foci of H, (b) the equations of the directrices of H. 16 9 = 1. 13. The ellipse E has equation + = 1. a b The line l1 is a tangent to E at the point P (a cos θ, bsin θ ). () [FP3 June 01Qn 1] (a) Using calculus, show that an equation for l1 is The circle C has equation cos sin + = 1. a b + = a. The line l is a tangent to C at the point Q (a cos θ, a sin θ ). (b) Find an equation for the line l. Given that l1 and l meet at the point R, (c) find, in terms of a, b and θ, the coordinates of R. (d) Find the locus of R, as θ varies. () () [FP3 June 01Qn 6] 14. The hperbola H has foci at (5, 0) and ( 5, 0) and directrices with equations = 9 5 and = 9. 5 Find a cartesian equation for H. (7) [FP3 June 013_R Qn 1] kumarmaths.weebl.com 10
15. The point P lies on the ellipse E with equation 1 36 9 N is the foot of the perpendicular from point P to the line = 8. M is the midpoint of PN. (a) Sketch the graph of the ellipse E, showing also the line = 8 and a possible position for the line PN. (1) (b) Find an equation of the locus of M as P moves around the ellipse. (c) Show that this locus is a circle and state its centre and radius. 16. A hperbola H has equation [FP3 June 013_R Qn 3] 1, where a is a positive constant. a 5 The foci of H are at the points with coordinates (13, 0) and ( 13, 0). Find (a) the value of the constant a, (b) the equations of the directrices of H. 17. The ellipse E has equation [FP3 June 013 Qn 1] 1, a > b > 0 a b The line l is a normal to E at a point P(acos θ, bsin θ), 0 < θ < (a) Using calculus, show that an equation for l is asin θ bcos θ = (a b ) sin θ cos θ The line l meets the -ais at A and the -ais at B. (b) Show that the area of the triangle OAB, where O is the origin, ma be written as ksin θ, giving the value of the constant k in terms of a and b. (c) Find, in terms of a and b, the eact coordinates of the point P, for which the area of the triangle OAB is a maimum. [FP3 June 013 Qn 7] kumarmaths.weebl.com 11
18. The ellipse E has equation + 9 = 9 The point P(a cos θ, b sin θ) is a general point on the ellipse E. (a) Write down the value of a and the value of b. The line L is a tangent to E at the point P. (1) (b) Show that an equation of the line L is given b 3 sin θ + cos θ = 3 The line L meets the -ais at the point Q and meets the -ais at the point R. (c) Show that the area of the triangle OQR, where O is the origin, is given b where k is a constant to be found. The point M is the midpoint of QR. k cosec θ (d) Find a cartesian equation of the locus of M, giving our answer in the form = f(). [FP3 June 014_R Qn 5] 19. The points P(3cos α, sin α) and Q(3cos β, sin β), where α β, lie on the ellipse with equation (a) Show the equation of the chord PQ is 9 4 1 cos sin cos 3 (b) Write down the coordinates of the mid-point of PQ. Given that the gradient, m, of the chord PQ is a constant, (1) (c) show that the centre of the chord lies on a line = k epressing k in terms of m. [FP3 June 014 Qn 6] kumarmaths.weebl.com 1
0. The hperbola H is given b the equation = 1 (a) Write down the equations of the two asmptotes of H. (b) Show that an equation of the tangent to H at the point P (cosh t, sinh t) is sinh t = cosh t 1. The tangent at P meets the asmptotes of H at the points Q and R. (1) (c) Show that P is the midpoint of QR. (d) Show that the area of the triangle OQR, where O is the origin, is independent of t. 1. The ellipse E has equation + 4 = 4 (a) (i) Find the coordinates of the foci, F1 and F, of E. (ii) Write down the equations of the directrices of E. (b) Given that the point P lies on the ellipse, show that [FP3 June 015 Qn 6] PF1 + PF = 4. A chord of an ellipse is a line segment joining two points on the ellipse. The set of midpoints of the parallel chords of E with gradient m, where m is a constant, lie on a straight line l. (c) Find an equation of l. (6) [FP3 June 015 Qn 8]. The hperbola H has equation 1. 16 9 The point P (4 sec θ, 3 tan θ), 0 < θ < lies on H. (a) Show that an equation of the normal to H at the point P is 3 + 4 sin θ = 5 tan θ. The line l is the directri of H for which > 0. The normal to H at P crosses the line l at the point Q. Given that θ = 4, (b) find the coordinate of Q, giving our answer in the form a + b numbers to be found., where a and b are rational (6) [FP3 June 016 Qn 5] kumarmaths.weebl.com 13
3. The ellipse E has equation 1 36 5 π The line l is the normal to E at the point P (6cos θ, 5sin θ), where 0 < θ < (a) Use calculus to show that an equation of l is The line l meets the -ais at the point Q. 6sin θ 5cos θ = 11sin θ cos θ The point R is the foot of the perpendicular from P to the -ais. (b) Show that OQ OR = e, where e is the eccentricit of the ellipse E. 4. The line with equation = 9 is a directri of an ellipse with equation 1 a 8 where a is a positive constant. Find the two possible eact values of the constant a. 5. The hperbola H has equation [FP3 June 017 Qn ] (6) [F3 IAL June 014 Qn ] 1 16 4 The line l is a tangent to H at the point P (4 cosh α, sinh α), where α is a constant, α 0. (a) Using calculus, show that an equation for l is sinh α cosh α + 4 = 0 The line l cuts the -ais at the point A. (b) Find the coordinates of A in terms of α. The point B has coordinates (0, 10 sinh α) and the point S is the focus of H for which > 0. (c) Show that the line segment AS is perpendicular to the line segment BS. () [F3 IAL June 014 Qn 6] kumarmaths.weebl.com 14
6. The hperbola H has equation 1 a b where a and b are positive constants. 1 The hperbola H has eccentricit and passes through the point (1, 5). 4 Find (a) the value of a and the value of b, (a) the coordinates of the foci of H 7. The ellipse E has equation 1 5 9 The line L has equation = m + c, where m and c are constants. Given that L is a tangent to E, (a) show that c 5m = 9 (b) find the equations of the tangents to E which pass through the points (3, 4). 8. An ellipse has equation (1) [F3 IAL June 015 Qn ] [F3 IAL June 015 Qn 5] 5 4 1 The point P lies on the ellipse and has coordinates (5 cos θ, sin θ), 0 < θ < The line L is a normal to the ellipse at the point P. (a) Show that an equation for L is 5 sin θ cos θ = 1 sin θ cos θ Given that the line L crosses the -ais at the point Q and that M is the midpoint of PQ, (b) find the eact area of triangle OPM, where O is the origin, giving our answer as a multiple of sin θ. (6) [F3 IAL June 016 Qn ] kumarmaths.weebl.com 15
9. The hperbola H has equation a and the ellipse E has equation a b 1 b 1 where a > b > 0 π The line l is a tangent to hperbola H at the point P (a sec θ, b tan θ), where 0 < θ < (a) Using calculus, show that an equation for l is b sec θ a tan θ = ab Given that the point F is the focus of ellipse E for which > 0 and that the line l passes through F, (b) show that l is parallel to the line = [F3 IAL June 017 Qn 6] kumarmaths.weebl.com 16