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Conics Merit - Excellence Question 1 A cross-section of a parabolic reflector is shown in the figure. A bulb is located at the focus. • (a) Find an equation for the parabola in parametric form relative to the axes shown. • (b) Find the position of the bulb relative to the axes shown. • General form of the equation: y = kx 2 Substitute for x and y y = kx 2 20 = 16k k = 25 2 y = 25x 2 y = 4 ( 6.25 ) x 2 Þ a = 6.25 Parametric form of the equation a = 6.25 x = at , y = 2at 2 x = 6.25t , y = 12.5t 2 Position of the bulb (6.25, 0) Question 2 The greatest and least distances from a seat on an elliptical ride at a fair ground to a focal point of the ellipse are 18 m and 2 m respectively. • (a) Find the equation of the locus of the seat relative to the axes shown. • (b) How high is the seat from the ground when the seat is in the position shown? Put the information on the diagram 2m 18m c = 8, a = 10 2m 18m c = 8, a = 10 2m 18m a =b +c 2 2 Þb=6 2 Equation is 2m y - 8) x ( + =1 2 2 10 6 2 18m 2 Substitute x = 5 2m 18m y - 8) 5 ( + =1 2 2 10 6 y = 8 + 3 3 = 13.2m 2 2 Question 3 A ship at position S receives radio signals from two radio stations located at A and B which are 640 km apart. The ship’s radio operator observes that the signals he receives are 1200 microseconds apart. Assuming that radio signals travel at a speed of 300 m/microsecond. • (a) Choosing a suitable coordinate system, write down an equation that describes the path of the ship. • (b) If the ship is due north of B, how far off the coastline is the ship? We are dealing with a hyperbola and will assume distance A – B is constant It could be two possibilities but the picture indicates that distance to A is greater c = 320 d1 - d2 = 1200 ´ 300m = 360km = 2a a = 180 The situation tells us that the locus is a hyperbola c = 320 a = 180 c2 = a2 + b2 b = 70000 x2 y2 =1 2 180 70000 If the ship is due north of B, how far off the coastline is the ship? (Assume A-B are West-East) x2 y2 =1 2 180 70000 x = c = 320 320 y =1 2 180 70000 y = 389km 2 2 Question 4 • Find the equation of the tangent to the curve 4x - 9y = 6 2 • at the point (2, 1). 2 X = 2, y = 1 4x - 9y = 6 2 2 dy 8x -18y = 0 dx dy 4 x 8 = = dx 9y 9 y = mx + c 8 1= ´2+c 9 7 c=9 4 7 y= x9 9 Question 5 • John shoots apples from his ‘potato canon’ over the edge of a cliff as shown on the diagram. He stands 6 m from the cliff edge and his apples reach a height of 4 m as shown. The path of his apples forms a parabola. • (a) Find an equation for the parabola relative to the axes shown. • (b) Find the height, h, of the cliff above the river bank if his apple lands on the edge of the river bank which is 15 m from the face of the cliff. y = kx 2 + 4 x = -6, y = 0 0 = 36k + 4 1 k =9 1 2 y=- x +4 9 (b) Find the height, h, of the cliff above the river bank if his apple lands on the edge of the river bank which is 15 m from the face of the cliff. x = 15 1 2 y=- x +4 9 1 y = - ´15 2 + 4 9 y = -21 h = 21 Question 6 • A tank, as shown, for storing petrol has an elliptical cross-section. The tank is 6 m wide and 4 m high. • (a) Find the equation of the cross-section relative to the axes shown. • (b) Find the equation of the cross-section (in parametric form) relative to the axes shown. • (c) Find the depth, d, of the tank 0.8 m from the edge. Find the equation of the cross-section relative to the axes shown. x y + 2 =1 2 3 2 2 2 Find the equation of the cross-section (in parametric form) relative to the axes shown x y + 2 =1 2 3 2 x = 3cosq 2 2 y = 2 sin q Find the depth, d, of the tank 0.8 m from the edge. x y + 2 =1 2 3 2 x = 2.2 2 2 y = 1.36m d = 1.36 + 2 = 3.36m Question 7 • An spy plane flies between two towns A and B which are 420 km apart, so that it is always 60 km closer to B than to A. • (a) Choosing a suitable coordinate system, write down an equation that describes the path of the spy plane. • (b) If the plane is due south of B, how far is the plane from town A? Draw a diagram d1 d2 Hyperbola “always 60 km closer to B than to A” d1 d2 2a = 60 a = 30 c = 210 c2 = a2 + b2 b = 43200 2 x y =1 900 43200 2 2 If the plane is due south of B, how far is the plane from town A? Assume AB are West-East d1 x = c = 210 d2 210 y =1 900 43200 y = -1440km 2 2 If the plane is due south of B, how far is the plane from town A? Assume AB are West-East d1 d2 210 y =1 900 43200 y = -1440km 2 d A = 420 2 +1440 2 = 1500km 2 Question 8 • A parabolic reflector has measurements as shown on the diagram. • (a) Find an equation for the parabola relative to the axes shown. • (b) If a light source is placed at the focus point, the reflector will produce a beam of parallel rays. Find the position of a light source if its beam is to produce parallel rays. y = k ( x -100 ) 2 400 2 = k ( 500 -100 ) k = 400 y = 400 ( x -100 ) 2 Find the position of a light source if its beam is to produce parallel rays. The light source is 100mm from the vertex. y 2 = 400 ( x -100 ) 4a = 400 a = 100 Question 9 • A lake is in the shape of an ellipse. The lake is 2 km wide and 10 km long. George’s house is situated 1.5 km from the eastern end of the lake as shown. • (a) Find the equation of the cross-section (in parametric form) relative to the axes shown. • (b) If he rows across the lake to Tom’s house and back every morning, how far does he row every morning in total. x = 5 cosq y = sin q If he rows across the lake to Tom’s house and back every morning, how far does he row every morning in total. x 2 y2 + =1 25 1 x = 3.5 y = 0.714 distance = 2.86 km Question 10 • Suppose a gun is heard first by one observer and then 1.5 seconds later by a second observer in another location a distance of 3 km from the first observer. Assume that the gun and the ear levels of the observers are all in the same horizontal plane and that the speed of sound at the time was 340 m/s. • (a) Choosing a suitable coordinate system, write down an equation that describes the possible position of the gun. • (b) If the gun is due north of the first observer, how far is the gun from the second observer? d2 - d1 = 1.5 ´ 340 1.5 = 0.255 + b 2 = 0.51km b 2 = 2.184975 2a = 51 a = 0.255 c = 1.5 2 d2 d1 2 1.5 = 0.255 + b 2 a = 0.255 x y =1 2 0.255 2.184975 d2 2 2 2 b 2 = 2.184975 d1 2 If the gun is due north of the first observer, how far is the gun from the second observer? x y =1 2 0.255 2.184975 d2 x = 1.5 2 2 y = 8.57 d = 32 + 8.57 2 = 9.1km d1 Question 11 • A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. • (a) Find an equation for the parabola relative to the axes shown. • (b) Find the diameter of the opening (CD), 12 cm from the vertex. Latus Rectum is 10 y =10x 2 Find the diameter of the opening (CD), 12 cm from the vertex. y =10x 2 x =12 y =10.95 Diameter = 21.9cm Question 12 • A semi-elliptic arch spans a motorway 50 m wide. • (a) Find an equation for the ellipse in parametric form relative to the axes shown. x 2 y2 + 2 =1 2 25 b x = 15, y = 14 2 2 15 14 + 2 =1 2 25 b b 2 = 306.25 x y + =1 2 25 306.25 2 2 (b) How high is the arch if a centre section of the highway 30 m wide has a minimum clearance of 14 metres. x y + = 1 Þ b = 17.5m 2 25 306.25 2 2 Question 13 An aircraft flies between two radio beacons A and B, which are 100 km apart, so that it is always 40 m closer to A than to B. • (a) Choosing a suitable coordinate system, write down an equation that describes the path of the aircraft. • (b) What approximate path would the aircraft be flying when it is 1000 km from A? c = 50 2a = d1 - d2 = 0.04 a = 0.02 c2 = a2 + b2 d1 d2 A 100 km b = 2500 2 B x2 y2 =1 2 0.02 2500 d1 d2 A 100 km B What approximate path would the aircraft be flying when it is 1000 km from A? d1 d2 A 100 km x2 y2 =1 2 0.02 2500 50 y=± x 0.02 y = ±2500x B Question 14 Find the equation of the tangent to the curve y2 - 2y - 4x + 4 = 0 • at the point (1, 2) Question 14 y 2 - 2y - 4x + 4 = 0 dy -4 + ( 2y - 2 ) = 0 dx dy 2 2 = = =2 dx y -1 2 -1 Question 14 y 2 - 2y - 4x + 4 = 0 dy -4 + ( 2y - 2 ) = 0 dx dy 2 2 = = =2 dx y -1 2 -1 x = 1, y = 2 2 = 2+c y = 2x