Deformity 9y ^ 2 = x ^ 3, at the point of deformity where the normal axis intersects similar builds.
To understand the deformity and the point where the normal axis intersects, let's break down the equation and analyze it step by step.
The equation given is "9y^2 = x^3." It is a mathematical equation in which the variables x and y are related according to the equation.
To find the point of deformity and where the normal axis intersects, we can start by examining the equation and its graphical representation.
The equation represents a curve that can be graphed in a coordinate plane. By plotting various points, we can see the shape of the curve and its characteristics.
First, let's rearrange the equation to isolate y:
9y^2 = x^3
Divide both sides by 9:
y^2 = (x^3)/9
Now, take the square root of both sides:
y = ±√((x^3)/9)
This equation gives us the relationship between x and y. However, it is not enough to find the specific point of deformity and the intersection with the normal axis.
To find these points, we need to differentiate the equation with respect to x. Differentiation helps find the slope of the curve, which is crucial in determining normal lines and points of deformity.
Differentiating the equation y = ±√((x^3)/9) with respect to x gives:
dy/dx = ±(3x^2)/(2√(9x))
Now, let's focus on the point of deformity. At the point of deformity, the slope is zero. So, we set dy/dx to zero and solve for x:
±(3x^2)/(2√(9x)) = 0
Since the term on the left side cannot be zero, we can conclude that there is no point of deformity in this equation.
Moving on to the intersection with the normal axis. The normal axis is the y-axis, where x = 0.
Substituting x = 0 into the equation y = ±√((x^3)/9), we get:
y = ±√((0^3)/9)
y = 0
Therefore, the curve intersects the y-axis at y = 0, which is the origin.
In summary, the given equation represents a curve with no point of deformity. The curve intersects the y-axis at the origin, where y = 0.