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INDEX
- Cur ve Tr aci ng
Curve Tracing
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Whose equation is in Cartesian form. 1. Symmetry (i) A curve is symmetrical symmetrical a bout x-axis if the equation remains the same by replacing y by –y. here y should have even powers only. For example y2 = 4ax. (ii) It is symmetrical about y-axis if i t contains only even powers powers of x For example x2 = 4ay. (iii )If on interchanging x and y, y, the equation remains the same then the curve is symmetrical about the line. y = x, For example example x3 + y3 = 3axy
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IIT JEE 2013 SOLUTION PAPER PA PER 1 Counter (iv) A curve is symmetrical in the opposite quadrants if its equation remains the same when x and y replaced by –x and –y. For example y = x 3 2. (a) Curve through Origin The curve passes through the origin, if the equation does not contain constant term. For example the curve y 2 = 4ax passes through the origin. 2 (b)Tangents at the origin:
To know the nature of a multiple point it necessary to find the tangent at that point. The equation of the tangent at t he origin can be obtained by equating to zero, the lowest degree term in the equation of the curve. 3. The points of intersection with the axes (i) By putting y= 0 in the equation of the curve we get the co-ordinates of the point of intersection with the x –axis.
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(ii) By putting x = 0 in the equation of the curve; the ordinate of the point of intersection with the the y-axis is obtained obtained by solving the new equation. 4. Regions in which the curve does not lie. lie . If the value of y is imaginary for certain value of x then the curve does not exist for such values. Example. y2 = 4x Example. a2x2 = y3(2a - y). For negative value of x, y is imaginary so there is no curve is second and third quadrant. 1. 2.
(i) For y> 2a x is imaginary so there is no curve in second and third quadrant (ii) For negative values, of y, x is imaginary. There is no curve in 3rd and 4th quadrant.
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5. Asymptotes are the tangents to the curve at infinity.
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(a) Asymptote parallel to the x-axis x-axis is obtained by equating to zero, the coefficient of the highest power of x.
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For example yx2 - 4x2 + x + 2 = 0
(y - 4)x2 + x + 2 = 0 The coefficient of the highest power of x i.e. x2 is y - 4 = 0 y - 4 = 0 is the asymptote parallel to the x axis. http://maths4i itj ee.page.tl /C ur ve- Tr aci ng.htm
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(b) Asymptote parallel to the y-axis is obtained to zero, the coefficient of highest power of y. For example xy3 - 2y3 + y2 + x2 + 2 = 0 (x-2)y3 + y2 + x2 + 2 = 0 The coefficient of the highest power of y. i.e. y3 is x -2. X -2 = 0 is the asymptote parallel to y-axis. 6. Tangent. Put 0=dx/dy for the points where tangent is parallel to the x-axis. For example x2 + y2 - 4x + 4y -1 = 0 ………(1) 2x +2ydxdy- 4 + 4dxdy= 0 (2y+4) dx/dy= 4 - 2x or dx/dy=4224+−yx Now dxdy= 0. 4-2x = 0 or x = 2 Putting x =2 in (1), we get y2 + 4y -5 = 0 y= 1,- 5 The tangents are parallel to x-axis at the points (2,1) and (2, -5). 7. Table. Prepare a table for certain values of x and y and draw the curve passing through them. For example y2 = 4x +4 x
-1
0
1
2
3
Y
0
±2
±22
±23
±4
Example. Trace the curve y 2( 2a – x ) = x 3 Solution: y 2= x3/( 2a – x ) ……..(1) (i) Origin: Equation does not contain any constant term; therefore, it passes through origin. (ii) Symmetrical about x-axis: Equation contains only even powers of y; therefore, it is symmetrical about x-axis. (iii)Tangent at the origin: Equation of the tangent is obtained by equating to zero the lowest degree terms in the equation (1). 2ay2 - xy2 = x3 Equation of tangent:
2ay2 = 0 → y2 = 0 (v) Asymptote parallel to y-axis: Equation of asymptote is obtained by equating the coefficient of lowest degree of y. 2ay2 - xy2 = x3 or (2a-x)y2 = x3 Eq. Of asymptote is 2a-x = 0 or x = 2a. (vi) Region of absence of curve: y2 becomes negative on putting x>2a or x<0, therefore, the curve does not exist for x<0 and x>2a.
Some more Examples
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Example. Trace the witch of Agnesi xy 2 = a2(a-x).
Example. Trace strophoid
x (x2 + y2 ) = a (x 2 - y 2 )
Example. Trace the Folium of Descart es x 3 + y3 = 3axy
Example. Trace y2( a 2 + x2 ) = x 2( a 2 - x 2 )
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