Preliminary Mathematics Extension 1

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1 Phone: (0) Web: dc.edu.au 018 HIGHER SCHOOL CERTIFICATE COURSE MATERIALS Preliminary Mathematics Extension 1 Parametric Equations Term 1 Week 1 Name. Class day and time Teacher name...

2 Term 1 Week 1 1 Term 1 Week 1 Theory CARTESIAN REPRESENTATION OF THE PARABOLA x = ±4ay (REVISION): A parabola is defined as the locus of all points that are equidistant from a fixed point and a given line. The point is known as the focus and the line is known as the directrix. The parabola with focus (0, a) and directrix y = a has Cartesian equation x = 4ay. The parabola with focus (0, a) and directrix y = a has Cartesian equation x = 4ay. EXAMPLE: Find the locus of a point P that moves so that its distance from the point (0, ) is the same as its distance from the line y =. Dux College 018 All rights reserved. T: (0)

3 Term 1 Week 1 SOLUTION: Let the point be P(x, y) Distance from P to (0, ) is given by (x 0) + (y ) = x + (y ) Distance from P to y = is given by 0x+1y+ = 0 +1 y+ = y + 1 x + (y ) = y + x + (y ) = (y + ) x + y 4y + 4 = y + 4y + 4 x = 8y The locus is x = 8y. PARAMETRIC EQUATION OF THE PARABOLA x = ±4ay The parabola x = 4ay can be represented by the parametric equations: x = at and y = at where t is known as a parameter. Therefore P(ap, ap ) would represent a general point on the parabola, and Q(aq, aq ) would represent another point on the parabola. This means that substituting in any given value for the parameter will give us exactly one point on the parabola, and the locus of all such points is the parabola. For example, consider the parabola y = x, or x = 4 ( 1 4 ) y (i.e. a = 1 4 ) If we let t =, we obtain the point ( ( 1 4 ) (), (1 4 ) () ) = (1, 1) If we let t = 4, we obtain the point ( ( 1 4 ) ( 4), (1 4 ) ( 4) ) = (, 4) The use of parametric representations allows important properties of the parabola and the equations of related curves (e.g. tangents, normals) to be expressed as functions of one parameter, t. This is helpful because it simplifies the algebra involved. In the following sections, both the parametric and Cartesian representations are used to derive the equations of tangents, normals and chords. Most questions will ask you to derive one or more of these equations in the first step, so that the results can be used to prove further properties. Therefore it is useful to know both the derivations and the results. These can be learnt simply through practicing on sample questions. There is no need to rote learn them. TANGENTS TO THE PARABOLA x = 4ay Dux College 018 All rights reserved. T: (0)

4 Term 1 Week Cartesian Representation Let P(x 1, y 1 ) be a point on the parabola x = 4ay. x = 4ay Differentiating both sides with respect to x, x = 4a. dy dx dy dx = x a at the point P(x 1, y 1 ), dy dx = x 1 a So the tangent is given by y y 1 = x 1 a (x x 1) ay ay 1 = xx 1 x 1 xx 1 = ay ay 1 + x 1 = ay ay 1 + 4ay 1 as (x 1, y 1 ) lies on x = 4ay. xx 1 = a(y + y 1 ) The tangent to the parabola x = 4ay at a point P(x 1, y 1 ) is given by xx 1 = a(y + y 1 ) Dux College 018 All rights reserved. T: (0)

5 Term 1 Week 1 4. Parametric Representation Let P(ap, ap ) be a point on the parabola x = 4ay with parameter p. x = 4ay Differentiating both sides with respect to x, x = 4a. dy dx dy dx = x a at the point P(ap, ap ), dy dx = ap a = p So the tangent is given by y ap = p(x ap) y ap = px ap y = px ap The tangent to the parabola x = 4ay at a point P(ap, ap ) is given by y = px ap Dux College 018 All rights reserved. T: (0)

6 Term 1 Week 1 5 NORMALS TO THE PARABOLA x = 4ay 1. Cartesian Representation Let P(x 1, y 1 ) be a point on the parabola x = 4ay. x = 4ay Differentiating both sides with respect to x, x = 4a. dy dx dy dx = x a at the point P(x 1, y 1 ), the gradient of the tangent to the curve is given by dy dx = x 1 a the gradient of the normal is a x 1 the normal to the parabola x = 4ay at a point P(x 1, y 1 ) is given by y y 1 = a x 1 (x x 1 ) Dux College 018 All rights reserved. T: (0)

7 Term 1 Week 1 6. Parametric Representation Let P(ap, ap ) be a point with parameter p on the parabola x = 4ay. x = 4ay Differentiating both sides with respect to x, x = 4a. dy dx dy dx = x a at the point P(ap, ap ), the gradient of the tangent to the curve is given by dy dx = ap a = p the gradient of the normal is 1 p So the normal is given by y ap = 1 (x ap) p py ap 3 = x + ap x + py = ap 3 + ap the normal to the parabola x = 4ay at a point P(ap, ap ) is given by x + py = ap 3 + ap Dux College 018 All rights reserved. T: (0)

8 Term 1 Week 1 7 INTERSECTION OF TANGE NTS AND NORMALS OF THE PARABOLA x = 4ay Many problems require you to find the intersection of the tangents or normals at point P(ap, ap ) and Q(aq, aq ), and then to prove some property involving this intersection. Thus is it worthwhile to know both what the point is, and how to derive it. 1. Intersection of Tangents Let the intersection be T Equation of the tangent at P: y = px ap (1) Equation of the tangent at Q: y = qx aq () Solving (1) and () simultaneously, px ap = qx aq px qx = ap aq (p q)x = a(p + q)(p q) x = a(p + q) as p q and so p q 0 y = px ap = ap(p + q) ap = ap(p + q p) = apq T[a(p + q), apq] is the intersection of the tangents at points P and Q.. Intersection of Normals Dux College 018 All rights reserved. T: (0)

9 Term 1 Week 1 8 Let the intersection be T Equation of normal at P: x + py = ap 3 + ap (1) Equation of normal at Q: x + qy = aq 3 + aq () Subtracting () from (1): py qy = ap 3 + ap aq 3 aq (p q)y = a(p 3 q 3 + (p q)) = a(p q)(p + pq + q + ) y = a(p + pq + q + ) as p q and so p q 0 x = ap 3 + ap py = ap 3 + ap ap(p + pq + q + ) = ap 3 + ap ap 3 ap q apq ap = ap q apq = apq(p + q) T[ apq(p + q), a(p + pq + q + )] is the intersection of the normals at points P and Q. Dux College 018 All rights reserved. T: (0)

10 Term 1 Week 1 Homework Term 1 Week By using differentiation, find the equation of the tangent to the parabola at the indicated points: a) x = t, b) x = 4t, c) x = t, d) x = at, e) x 4y y = t at the point where t = 1 y = t at the point where 1 t 1 t = y = at the point where t = 4 y = at at the point where t = 3 = at the point (,1 ) f) x 1 = 8y at the point, g) x 6y = at the point ( 6,6 ) = at the point ( x ) h) x 4ay 1, y 1 Dux College 018 All rights reserved. T: (0)

11 Term 1 Week (i) Find the equation of the tangent to the parabola = ( ) at the point 4t,t x 8y (ii) Hence determine all tangents to the parabola that pass through the point ( 1, 1) 3. P ( p, p ) and ( ( 1 1 ), ( ) ) Q are two variable points on the parabola x = 4y. The tangents at P and p p Q intersect at a point T. (i) Find the equation of the tangent to the parabola at P. (ii) Determine the coordinates of T. (iii) Hence find the Cartesian equation of the locus of T. Dux College 018 All rights reserved. T: (0)

12 Term 1 Week The line ax +by = 1 is tangent to the parabola x 4y =. Find the conditions on a and b. P is a variable point on the parabola x 4ay 5. ( ap, ap ) =. The tangent at P intersects the x-axis at A and the y-axis at B. C is the fourth vertex of rectangle OACB. (i) Find the coordinates of C in terms of p. (ii) Hence show that the locus of C is a parabola and state its vertex and focus. Dux College 018 All rights reserved. T: (0)

13 Term 1 Week 1 1 P is a variable point on the parabola x 4ay 6. ( ap, ap ) =. T is the foot of the perpendicular drawn from the focus to the tangent at P. Find the Cartesian equation of the locus of T. Dux College 018 All rights reserved. T: (0)

14 Term 1 Week By using differentiation, find the equation of the normal to the parabola at the indicated points: a) x = t, b) x = 6t, c) x = t, d) x = at, e) x 4y y = t at the point where t = y = 3t at the point where t = 4 1 t y = at the point where t = 1 y = at at the point where t = p = at the point (,1 ) = at the point ( 1, 1) = at the point ( 6,3 ) f) x y g) x 1 y 1 4 = at the point ( x ) h) x y 1, y 1 Dux College 018 All rights reserved. T: (0)

15 Term 1 Week (i) Find the equation of the normal to the parabola x = at, y = at at the point where t = p. (ii) The normal intersects the x-axis at A and the y-axis at B. Find the coordinates of A and B. (iii) Hence determine the area of AOB Dux College 018 All rights reserved. T: (0)

16 Term 1 Week (i) Find the equation of the parabola that is symmetrical about the y-axis and passes through the points 4,4. ( 1,1) and ( ) (ii) Find the normal to the parabola at the point ( 1,1). Dux College 018 All rights reserved. T: (0)

17 P is a variable point on the parabola x 4ay 10. ( ap, ap ) Term 1 Week 1 16 =. The normal at P intersects the y-axis at T. M is the midpoint of PT. (i) Find the coordinates of T. (ii) Hence find the coordinates of M and determine the Cartesian equation of the locus of M. End of Homework Dux College 018 All rights reserved. T: (0)

Name. Center axis. Introduction to Conic Sections

Name. Center axis. Introduction to Conic Sections Name Introduction to Conic Sections Center axis This introduction to conic sections is going to focus on what they some of the skills needed to work with their equations and graphs. year, we will only