Final answer:
The locus of points that is five inches from a given point and ten inches away from a given line that is five inches from the given point is a hyperbola.
Step-by-step explanation:
The locus of points that is five inches from a given point and ten inches away from a given line that is five inches from the given point is known as a hyperbola. A hyperbola is a type of conic section that consists of two distinctly curved branches. The distance between the center of the hyperbola and each branch is constant, known as the distance between the branches.
To understand this concept, let's consider an example. Suppose we have a point P(0,0) and a line L with equation y = 5. The locus of points that is 5 inches from P and 10 inches away from line L would be a hyperbola with the center at (0,0) and a vertical transverse axis. The equation of this hyperbola would be:
x^2 / a^2 - y^2 / b^2 = 1
Where 'a' is the distance between the center and each vertex of the hyperbola, and 'b' is the distance between the center and each branch of the hyperbola. In this case, a = 5 (since the distance from P to the vertex is 5) and b = 10 (since the distance from P to each branch is 10).