Tangents at the Origin
Let the equation of the curve passing through the origin be
(a_{1}x + a_{2}y) + (b_{1}x^{2} + b_{2}xy + b_{3}y^{2}) + …. + (l_{1}x^{n} + … + l_{n}y^{n}) = 0 (1)
Let P (x, y) be any point on the curve. The slope of the chord OP is . Thus the equation of the chord OP is
Y = .
As x0, the chord OP becomes the tangent at O and so the equation of the tangent at O is
Y = mX, where m = lim_{x0} . (2)
Dividing (1) by x, we obtain
Taking the limit as x0 and using (2), we obtain
a_{1} + a_{2}m = 0
i.e. a_{1} + a_{2} Y/X = 0 (∵ Y = mX)
i.e. a_{1}X + a_{2}Y = 0.
Thus the equation of the tangent at the origin may be taken as
a_{1}x + a_{2}y = 0.
This equation is same as the lowest degree terms in (1) when equated to zero.
If a_{1} = a_{2} = 0, then (1) becomes
(b_{1}x + b_{2}xy + b_{3}y^{2}) + (c_{1}x^{3} + c_{2}x^{2}y + c_{3}xy^{2} + c_{4}y^{3}) + …. = 0 (3)
Dividing the x^{2} and taking the limit as x0, we obtain
b_{1} + b_{2}m + b_{3}m^{2} = 0
Or, b_{1} + = 0 (∵ Y = mX)
Or, b_{1}X^{2} + b_{2}XY + b_{3}Y^{2} = 0.
We may write it as
b_{1} x^{2} + b_{2}xy + b_{3}y^{2} = 0. (4)
which represents a pair of tangents at this origin.
The equation (4) is same as the lowest degree terms in (3) when equated to zero.
Similarly, it can be shown that if a_{1} = a_{2} = 0 and b_{1} = b_{2} = b_{3} = 0, then c_{1}x^{3} + c_{2}x^{2}y + c_{3}xy^{2} + c_{4}y^{3} = 0 is the equation of the tangent at the origin. Hence we have the following:
Rule: The tangents at the origin are given by equating to zero the lowest degree terms in the equation of the given curve.
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