Coordinate goemetry in the (x, y) plane

Transcription

1 Coordinate goemetr in the (x, ) plane In this chapter ou will learn how to solve problems involving parametric equations.. You can define the coordinates of a point on a curve using parametric equations. In parametric equations, coordinates x and are expressed as x f(t) and g(t), where the variable t is a parameter. Example Draw the curve given b the parametric equations x t, t, for t. t x t 6 6 t 9 9 Draw a table to show values of t, x and. Choose values for t. Here t Work out the value of x and the value of for each value of t b substituting values of t into the parametric equations x t and t. e.g. for t : x t t () () So when t the curve passes through the point (, ). 6 6 x Plot the points ( 6, 9), (, ), (, ), (, ), (, ), (, ), (6, 9) and draw the graph through the points. 9

2 CHAPTER Example A curve has parametric equations x t, t. Find the cartesian equation of the curve. x t So t x A cartesian equation is an equation in terms of x and onl. To obtain the cartesian equation, eliminate t between the parametric equations x t and t. t Substitute into : x The cartesian equation is x. Rearrange x t for t. Divide each side b. Substitute t = x into t. Expand the brackets, so that x x x x Exercise A A curve is given b the parametric equations x t, t where t. Complete the table and draw a graph of the curve for t. t.. x t 8. t A curve is given b the parametric equations x t, t. Complete the table and draw a graph of the curve for t. t x t 6 t.8 Draw the curves given b these parametric equations: a x t, t for t b x t, t for t c x t, t( t) for t d x t, t t for t e x t, ( t)(t ) for t

3 Coordinate geometr in the (x, ) plane Find the cartesian equation of the curves given b these parametric equations: a x t, t b x t, t c x t, t, t d x t,, t t e x t, 9 t f x t, t(9 t) g x t, (t )(t ) h x, t, t t t t i x,, t, t j x,, t, t t t t t Show that the parametric equations: t i x t, t ii x,, t t t represent the same straight line.. You need to be able to use parametric equations to solve problems in coordinate geometr. Example parametric equations x t, t. The curve meets the x-axis at the points A and B. Find the coordinates of A and B. t Substitute t t So t x t Substitute t x () Substitute t x ( ) The coordinates of A and B are (, ) and (, ). A B x Find the values of t at A and B. The curve meets the x-axis when, so substitute into t and solve for t. Take the square root of each side. Remember there are two solutions when ou take a square root. Find the value of x at A and B. Substitute t and t into x t.

4 CHAPTER Example A curve has parametric equations x at, a(t 8), where a is a constant. The curve passes through the point (, ). Find the value of a. a(t 8) Substitute a(t 8) t 8 t 8 So t x at Substitute x and t a( ) So a (The parametric equations of the curve are x t and (t 8)). The curve passes through (, ), so there is a value of t for which x and. Find t. Substitute into a(t 8) and solve for t. Divide each side b a. Take the cube root of each side. 8. Find a. Substitute x and t into x at. Divide each side b. Example A curve is given parametricall b the equations x t, t. The line x meets the curve at A. Find the coordinates of A. x Substitute t t (t ) t So t Substitute t x t ( ) t ( ) 8 The coordinates of A are (, 8). Find the value of t at A. Solve the equations simultaneousl. Substitute x t and t into x. Factorise. Take the square root of each side. Find the coordinates of A. Substitute t into the parametric equations.

5 Coordinate geometr in the (x, ) plane Exercise B Find the coordinates of the point(s) where the following curves meet the x-axis: a x t, 6 t b x t, t 6 c x t, ( t)(t ) d x, (t (t ) t ) t e x, t 9, t t Find the coordinates of the point(s) where the following curves meet the -axis: a x t, t b x (t ),, t t c x t t, t(t ) d x 7 t,, t t e x t t,, t t t A curve has parametric equations x at, a(t ), where a is a constant. The curve passes through the point (, ). Find the value of a. A curve has parametric equations x b(t ), b( t ), where b is a constant. The curve passes through the point (, ). Find the value of b. A curve has parametric equations x p(t ), p(t 8), where p is a constant. The curve meets the x-axis at (, ) and the -axis at A. a Find the value of p. b Find the coordinates of A. A curve is given parametricall b the equations x qt, (t ), where q is a constant. The curve meets the x-axis at X and the -axis at Y. Given that Ox OY, where O is the origin, find the value of q. Find the coordinates of the points of intersection of the line with parametric equations x t, t and the line x. Find the coordinates of the points of intersection of the curve with parametric equations x t, (t ) and the line x. Find the values of t at the points of intersection of the line x with the parabola x t, t and give the coordinates of these points. Find the points of intersection of the parabola x t, t with the circle x 9x.

6 CHAPTER. You need to be able to convert parametric equations into a cartesian equation. Example 6 A curve has parametric equations x sin t, cos t. a Find the cartesian equation of the curve. b Draw a graph of the curve. a x sin t So sin t x cos t So cos t As sin t cos t, the cartesian equation of the curve is (x ) ( ) Eliminate t between the parametric equations x sin t and cos t. Remember sin tt cos tt.. Rearrange x sin t for sin t. Take from each side. Rearrange cos t for cos t. Add to each side. b x (, ) Square sin t and cos t so that sin t (x ) cos t ( ). Remember (x a) ( b) r is the equation of a circle centre (a, b) radius r. Compare (x ) ( ) to (x a) ( b) r. Here a, b and r. So (x ) ( ) is a circle centre (, ) and radius Example 7 A curve has parametric equations x sin t, sin t. Find the cartesian equation of the curve. Eliminate t between the parametric equations x sin t and sin t. sin t sin t cos t x cos t Remember sin t sint cos t. x sin t, so replace sin t b x in sint cos t.

7 Coordinate geometr in the (x, ) plane sin t cos t cos t sin t cos t x cos t ( ) x So x ( ) x Find cos t in terms of x. Rearrange sin t cos t for cos t. Take sin t from each side. x sin t, so replace sin t b x in cos t sin t. Take the square roof of each side. Replace cos t b ( ) x in x cos t. This equation can also be written in the form x ( x ). Exercise C A curve is given b the parametric equations x sin t, cos t. Complete the table and draw a graph of the curve for t. t x sin t.7.7 cos t.87.. A curve is given b the parametric equations x sin t, tan t, t. Draw a graph of the curve. Find the cartesian equation of the curves given b the following parametric equations: a x sin t, cos t c x cos t, sin t e x sin t, cos t g x cos t, cos t b x sin t, cos t d x cos t, sin t f x cos t, sin t h x sin t, tan t i x cos t, sec t j x cot t, cosec t A circle has parametric equations x sin t, cos t. a Find the cartesian equation of the circle. b Write down the radius and the coordinates of the centre of the circle. A circle has parametric equations x sin t, cos t. Find the radius and the coordinates of the centre of the circle.

8 CHAPTER. You need to be able to find the area under a curve given b parametric equations. The area under a graph is given b dx. B the chain rule dx d x. Example 8 A curve has parametric equations x t, t. Work out d x. So d x t d x d x d (t ) d t t d x t t t d x t t () () 6 6 Find d x. Substitute t. Work out d x. Here x t. d (t ) t d t t Simplif the expression so that t t t t Integrate, so that t t t t Work out t. Substitute t and t into t and subtract. Example 9 parametric equations x t, t( t), t. The curve meets the x-axis at x and x 9. The shaded region R is bounded b the curve and the x-axis. a Find the value of t when i x ii x 9 b Find the area of R. R 9 x 6

9 Coordinate geometr in the (x, ) plane a i x t t Substitute x into x t. Take the square root of each side. So t ii x t t 9 So t Substitute x 9 into x t. Take the square root of each side. 9. Ignore t as t. b Area of R 9 d x dx t( t) d x t( t) t (6t t ) t t t t t [() ()] [() () ] (8 8) ( ) 7 The area of R 7. Integrate parametricall. Change the limits of the integral. t when x t when x 9 Find d x. Substitute t( t). Work out d x. Here x t. d x d (t ) d t t Expand the brackets, so that t( t) t t t 6t t (6t t ) t 6t t t t t t Integrate each term, so that t t t t t t Work out t t. Substitute t and t into t t and subtract. 7

10 CHAPTER Exercise D The following curves are given parametricall. In each case, find an expression for d x in terms of t. a x t, t b x t t, t c x (t ), t d x 6 t, t, t e x t, 6t f x t, t, t g x t, t h x t, t i x 6 t,, t j x 6t, t t 6 7 A curve has parametric equations x t, t 8. Work out d x. A curve has parametric equations x t t, t. Work out d x. A curve has parametric equations x t, t t, t. Work out d x.. A curve has parametric equations x t t, t. Work out d x. A curve has parametric equations x 9t, t, t. a Show that d x a, where a is a constant to be found. b Work out d x. A curve has parametric equations x t, t, t. a Show that d x pt, where p is a constant to be found. b Work out 6 d x. 8 parametric equations x t, t, t. The shaded region R is bounded b the curve, the x-axis and the lines x and x 6. a Find the value of t when R ii x ii x 6 6 x b Find the area of R. 8

11 Coordinate geometr in the (x, ) plane 9 parametric equations x t, t( t), t. The shaded region R is bounded b the curve and the x-axis. Find the area of R. R x The region R is bounded b the curve with parametric equations x t, t, the x-axis and the lines x and x 8. a Find the value of t when i x ii x 8 b Find the area of R. Mixed exercise E parametric equations x cos t, sin t, t. a Find the coordinates of the points A and B. b The point C has parameter t. Find the exact 6 coordinates of C. c Find the cartesian equation of the curve. B C A x parametric equations x cos t, sin t, t. The curve is smmetrical about both axes. a Cop the diagram and label the points having parameters t, t, t and t. b Show that the cartesian equation of the curve is x ( x ). x A curve has parametric equations x sin t, cos t, t. a Find the cartesian equation of the curve. The curve cuts the x-axis at (a, ) and (b,). b Find the value of a and b. A curve has parametric equations x,, t. t ( t) ( t) Express t in terms of x. Hence show that the cartesian equation of the curve is x. x 9

12 CHAPTER A circle has parametric equations x sin t, cos t. a Find the cartesian equation of the circle. b Draw a sketch of the circle. c Find the exact coordinates of the points of intersection of the circle with the -axis. Find the cartesian equation of the line with parametric equations x t, t t, t. t A curve has parametric equations x t, t t, where t is a parameter. a Draw a graph of the curve for t. b Find the area of the finite region enclosed b the loop of the curve. A curve has parametric equations x t, t, where t. a Draw a graph of the curve. b Indicate on our graph where i t ii t iii t c Calculate the area of the finite region enclosed b the curve and the -axis. Find the area of the finite region bounded b the curve with parametric equations x t,, the x-axis and the lines x and x 8. t parametric equations x t, t( t), where t. The region R is bounded b the curve and the x-axis. a Show that d x 6t t. b Find the area of R. R 6 x Summar of ke points To find the cartesian equation of a curve given parametricall ou eliminate the parameter t between the parametric equations. To find the area under a curve given parametricall ou use d x.