A Text book of Differential Calculus With Numerous Worked Out Examples

Ganesh Prasad

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Y -) + 2xy(x' i — y-) + x* + y' 2 = 1 .
4. Write down the equations of the real asymptotes of the curves — (i) xy 3 + x 3 y = a 4 . (ii) xy' 2 = -la' 2 ('2a-x). (iii) x' 2 y' 2 = a' 2 (x' 2 — y' 2 ).
f A fuller treatment wil^be given in Vol. II.
ASYMPTOTES 59 45*. General Rule for finding Asymptotes from Polar Equation.
Let the equation of a curve be — =/(ti) ; then if a be a root of the equation /(0)=O, rsin(e-a)= f/ ^ ) .... (1) is an asymptote to the curve.
For, the vectorial angles of the p...oints of intersection of the secant r sin (0— -fi)=a with the curve are the roots of the equation 0(0)=a/(0)— sin (0-/3)=O.
Now when the secant is made to tend to (1), /3 and a must tend to a and, .. - ■ respectively, and both (fl)=0 tend to «, and consequently the corresponding radii vectores tend to oo.
Note 1. The first part of the above sentence will become quite clear to the reader if he notes that, according to a general theorem which he will study in Chapter X.

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