Differential And Integral Calculus

Cover Differential And Integral Calculus
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(a) When the conic intersects the x-axis.
In this case the quadratic expression ax 2 + bx + c has real factors, say, ax 2 + bx + c = a(x a) (x /3).
The conic (1) intersects the #-axis in the two points (a, 0) and (/3, 0), either one of which may be conveniently selected for the point Q.
The equation of any line QP through the first point is and the equation of any line through the second point, y = (*-/8)- (4') Either one of these equations, combined with (1), will effect the desired rationaliz
...ation.
(6) When the conic is an hyperbola.
This case occurs when the coefficient of x 2 is positive. The curve extends to infinity in two different directions, namely, the directions of the asymptotes. If one of the points at infinity on the curve be taken for the point (), the lines QP passing through this point are parallel to that 139. ] INTEGRATION BY RATIONALIZATION 235 asymptote which touches the curve at Q. The equations of the asymptotes are Accordingly the lines parallel to the one asymptote are y = Va x 4- 2, (-B) and those parallel to the other y = Va x + 2.


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