The strophoid is a special case of cissoid and is usually defined as a looping curve. It was first studied by Issac Barrow (October 1630 ? 4 May 1677) in 1670.

If only even powers of y occur in (1), i.e., if y is replaced by -y in (1), equation (1) remains unaltered or in or words f(x, -y) = f(x, y).

• The Folium of Descartes is an algebraic curve defined as, x³+y³-3xy=0.
• It is symmetrical about y = x.
• It forms a loop in first quadrant with a double point at origin and asymptote, x+y+a=0.

In an interval, a?x?b,

• If \(\frac{dy}{dx}>0, \) n curve is increasing in [a, b]
• If \(\frac{dy}{dx}< 0, \) n curve is increasing in [a, b]
• If \(\frac{d^2x}{dy^2}>0, \) n curve is convex and concave upward (holds water)
• If \(\frac{d^2x}{dy^2}>0, \) n curve is concave downward (spills water)

Oblique asymptotes are those which are neir parallel to x-axis nor y-axis and are given by y=mx+c where \(m=\lim_{x\to ?}?(\frac{x}{y})=\lim_{x\to ?}(y-mx).\)

Out of given options, x⁵+y⁵=5a² x² y represents curve symmetric about origin and can be observed as,

Oblique asymptotes are those which are neir parallel to x-axis nor y-axis and are given by y=mx+c where \(m=\lim_{x\to ?}?(\frac{x}{y})=\lim_{x\to ?}(y-mx).\)

Given: x³+ y³=3axy
To find tangent to curve at origin, we need to equate lowest degree term to 0.
Therefore, 3axy=0, which gives x=0 and y=0 as two tangents to curve at origin.

The curve of Cissoid was invented by Diocles in about 180 BC. The curve is given by,
Cartesian Equation: \(y^2=\frac{x^3}{(2a-x)} \)
Polar Equation: r=2a*tan?*sin?

The Lemniscate of Bernoulli was first described as a modification of an ellipse in 1694 by Jakob Bernoulli. It is represented as,
Cartesian Equation: (x²+y²)²=2a² (x²-y²)
Polar Equation: r²=2a²*cos2?