Curve Tracing in Cartesian coordinates.pptx
- 1. Curve Tracing in Cartesian coordinates Prof.Rootvesh Mehta, Indus University
- 2. Meaning of Tracing of Curves • Generally a curve is drawn by plotting a number of points and joining them by a smooth line. • If an approximate shape of the curve is sufficient for a given purpose then it is enough to study certain important characteristics. This purpose is served by curve tracing methods.
- 3. For tracing of curves we need to discuss following points • Symmetry • Origin and Tangents at the Origin • Intercepts • Asymptotes • Special points • Asymptote • Region
- 4. 1-Symmetry • (a) Symmetric about x-axis: If all the powers of y occurring in the equation are even then the curve is symmetrical about x-axis. • ( b) Symmetric about y-axis: If all the powers of x occurring in the equation are even then the curve is symmetrical about y-axis. • (c) Symmetric about both x- and y-axis: If only even powers of x and y appear in equation then the curve Is symmetrical about both axis. • (d) Symmetric about origin: If equation remains unchanged when x and y are replaced by – x and – y. • Remark: Symmetry about both axis is also symmetry about origin but not the converse (due to odd powers) • (e) Symmetric about the line y = x : A curve is symmetrical about the line y = x, if on interchanging x and y its equation does not change. • (f) Symmetric about y = – x : A curve is symmetrical about the line y = – x, if the equation of curve remains unchanged by putting x = – y and y = – x in equation.
- 5. • In short ,Whether the curve is symmetric about an axis or about other any line. If • F(x, y) = F(x,-y) => curve is symmetric about x-axis • F(-x, y) =F(x, y) => curve is symmetric about y-axis • F(-x,-y) =F(x, y) => curve is symmetric in opposite quadrants. • F(y, x) = F(x, y) curve is symmetric about y = x • F(-y,-x) =F(x, y) curve is symmetric about y =-x • Symmetric about origin: If equation remains unchanged when x and y are replaced by – x and – y.
- 6. 2-Origin 3-Tangents at the Origin • If there is no constant term in the equation then the curve passes through the origin otherwise not. • If F(0,0) = 0 => the curve is passing through origin. i.e. if the eqn. satisfies by the point(0,0) then it passes through the origin. • If the curve passes through the origin, then the tangents to the curve at the origin are obtained by equating to zero the lowest degree terms.
- 7. 4-Intercepts • (a) Intersection point with x- and y-axis: Putting y = 0 in the equation we can find points where the curve meets the x–axis. Similarly, putting x = 0 in the equation we can find the points where the curve meets y-axis • (b) Points of intersection: When curve is symmetric about the line y = ± x, the points of intersection are obtained by putting y = ± x in given equation of curve. • (c) Tangents at other points say (h, k) can be obtained by shifting the origin to these points (h, k) by the substitution x = x + h, y = y + k and calculating the tangents at origin in the new xy plane. • Ponits where tangents are parallel to coordinate axes:- The point where dy/dx = 0, the tangent is parallel to x-axis. And the point where dy/dx = ∞, the tangent is vertical i.e., parallel to y-axis.
- 8. 5-Asymptotes • Finding the asymptotes. • An asymptote is a line that is at a finite distance from (0, 0) and is tangential to the curve at infinity (i.e.) the curve approaches the line at infinity. • A tangent to the curve at infinity is called its asymptote. • Asymptotes parallel to co-ordinate axes. • To find asymptote parallel to X-axis Equate the coefficients of the highest degree terms in x with 0. • To find asymptote parallel to Y-axis Equate the coefficients of the highest degree terms in y with 0. • To find the asymptotes that are neither parallel to x-axis nor parallel to y axis (i.e.) oblique asymptotes, the following method is suggested.
- 9. Procedure for finding oblique asymptote • Let y=mx+c be the equation asymptote to the curve. • Form an nth degree polynomial of m by putting x=1,y=m in the highest degree term of given eqn.of curve. 1 Let ( ) ( ) be polynomials of terms of degree n and n-1 respectively. n n m and m Solve ( ) 0 for finding m. n m 1 ' ( ) Find c by the formula c= ( ) and substitute value of m and c in y=mx+c we get oblique asymptote. n n m m
- 10. 6-Study of special points on the curve • A point through which r branches of a curve pass is called a multiple point of order r and has r tangents at that point. multiple point(double point, triple point etc…) • In particular if two branches of the curve passes through it then it is known as Double point and at double point two tangents exists. • The double point may be classified as a) node b) cusp c) conjugate point as per nature of tangents. • If the tangents are: • (i) real and coincident then point is called a cusp • (ii) real and different then point is called a node • (iii)imaginary then point is called a conjugate point • For many curves the multiple point will be origin. But it is possible that the multiple point (a,b) other than origin may also exists and we will use following method for finding it.
- 12. 7-Regions (a) Region where the curve exists: It is obtained by solving y in terms of x or vice versa. Real horizontal region is defined by values of x for which y is defined. Real vertical region is defined by values of y for which x is defined (b) Region where the curve does not exist: This region is also called imaginary region, in this region y becomes imaginary for values of x or vice versa. That means , • Find the region in which the curve exists (i.e.) the curve is defined. The values of x for which y is defined gives the extent parallel to x-axis and the values of y for which x is defined gives the extent in a direction parallel to y-axis. • If y is imaginary for values of x in a certain region then the curve does not exist in that region. Similarly with respect to values of y
- 13. It is better to discuss various above points for solving problems in following sequence:- • Symmetry • Origin • Tangents at the origin • Intercepts • Special points • Asymptote • Region • Sign of derivatives (not necessary)
- 14. Ex. 1. Trace the curve Solution: 1. Symmetry: In the given equation y has even power only therefore curve is symmetry along x- axis. 2. Origin: Since equation of curve has constant term a so that it will not pass through the origin. 3. Asymptote: putting coefficient of highest degree of y equals to zero. i.e. x = 0 is an asymptote parallel to y axis. 4. Points: Putting y = 0 in equation of curve. x = a is the point where curve passes through on x axis. 2 2 ( ) xy a a x
- 15. 5. Region: For x > 0, y is imaginary. 04/18/2025
- 16. Ex.2. Solution: 1. Symmetry: Since even power of x exist in the given equation of curve, therefore it is symmetry along y-axis. 2. Origin: Since there is no any independent constant in the given equation therefore it passes through the origin. i.e. it satisfies x=0, y=0. 3. Asymptote: There is no any asymptote. 04/18/2025 2 2 3 (2 ) a x y a y
- 17. Ex.2. Solution: 4. Point: Putting x=0, we get y=0 and 2a. Thus the point passes through the y=0 and y=2a 5. Region: For y<0 and y>2a, x is imaginary. 04/18/2025 2 2 3 (2 ) a x y a y
- 18. 04/18/2025 2 2 3 (2 ) a x y a y
- 19. Ex.3. Solution: 1. Symmetry: Interchanging x and y the equation is unaltered, hence the curve is symmetrical with respect to the line y=x. 2. Origin: The curve passes through the origin. The tangents at the origin are obtained by equating to zero the lowest degree term in the equation i.e. 3axy=0. => xy=0, hence x=0, y=0. therefore x-axis and y-axis are tangents at the origin. 3. Asymptote: (a) it has no asymptotes parallel to the axes. 04/18/2025 3 3 3 x y axy
- 20. (b) Putting y = m and x = 1 in the third degree terms Point: Since the two tangents at the origin are real and distinct, therefore origin is a node. For intersection with x-axis, we put y = 0, 04/18/2025 3 3 3 1 , ( ) 0 1 m m m m 2 ' 2 3 3 3 , 3 1 m am a m am c m m m c a when m . . 0 y mx c y x a i e x y a is an asymptote
- 21. and for intersection with y-axis we put x = 0, Thus the curve meets the co-ordinate axes at (0,0) When y = x, x= 0 or x = 3a/2. The curve crosses the line y = x at and . 04/18/2025 3 3 , 2 2 a a 3 0, 0. x x 3 0, 0. y y 0,0
- 22. Region: Both x and y cannot be negative for otherwise the left hand side of the equation will be negative and right hand side will be positive. No portion of the curve, therefore lies in the third quadrant. 04/18/2025
- 23. 04/18/2025 3 3 3 x y axy