Question

Trace the curve ay^2 = x²(x - a).​

Answer 1

Answer:

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Step-by-step explanation:

1. Symmetry: The curve is symmetrical about X-axis only.  2. Origin: The equation of the curve does not have any constant term, so it passes through the origin and the tangent at the origin are y2 – 3x2 = 0 or y = ± √3x . The origin is a node.  3. Asymptote: The curve possesses asymptote parallel to Y-axis, viz. a + x = 0.  4. Points: Intersection with the X-axis y = 0 ⇒ x = 0, 0, 3a Thus the curve passes through (0, 0) and (3a, 0) Also from given equation y =  so that  Origin: As there is no constant term in the equation of the curve, it passes through the origin. The tangents at the origin are obtained by equating to zero the lowest degree term in the equation of curve. a(y2 – x2) = 0, i.e. y = ±x                (The tangent is a node) Asymptotes: From (1), a + x = 0, i.e. x = –a is an asymptote parallel to Y-axis. Points: The curve intersects X-axis at x2 (a – x) = 0, i.e. x = 0 and x = a (whereas it does not intersects Y-axis) Further    becomes infinite for (a – x) 1/2 (a + x) 3/2 = 0 i.e. at x = a, Y-axis is tangent at the point (a, 0)  Regions:  y has imaginanry values for x > a and from x = 0 to x = a, it first increases from O onwards than becomes zero at x = a  Again for –∞ < x < –a , y is imaginary Read more on Sarthaks.com - https://www.sarthaks.com/495575/trace-the-curve-y-2-a-x-x-2-3a-x