the family of curves ( x^2)( y^2)=c^2
The equation given, (x^2)(y^2) = c^2, represents a family of curves. Let's understand how to interpret this equation and the relationship between the variables.
In this equation, x and y are variables, while c is a constant. The constant c^2 determines the shape and position of the curves within the family.
To analyze the equation further, let's consider different scenarios when the constant c takes different values:
1. When c = 0:
In this case, the equation becomes x^2 * y^2 = 0^2, which simplifies to x^2 * y^2 = 0.
Here, only one curve exists, which is the coordinate axis (x = 0 or y = 0).
2. When c > 0:
Let's consider c = 1 for simplicity. In this case, the equation becomes x^2 * y^2 = 1^2.
To solve this equation and understand the shape of the curve, we can rewrite it as:
(y^2) = (1^2)/(x^2), or y^2 = 1/x^2.
Taking the square root of both sides, we get y = ±1/x.
This represents two curves symmetrically located around the coordinate axes. As the value of c increases, the curves get closer to the coordinate axes.
3. When c < 0:
Similar to the previous scenario, let's consider c = -1 for simplicity. The equation becomes x^2 * y^2 = (-1)^2.
Simplifying further, we have y^2 = 1/x^2, which results in y = ±1/x.
Again, this represents two curves symmetrically positioned around the coordinate axes. As c decreases (in absolute value), the curves move away from the coordinate axes.
In summary, the family of curves (x^2)(y^2) = c^2 consists of hyperbolas that are symmetrically situated around the coordinate axes. By adjusting the constant c, you can control the position and shape of the curves.