Answer:
(x/9)² +(y/3)² = 1 . . . P = (x, y)
Step-by-step explanation:
You want the equation of the locus of a point P that is 3 cm from the x-axis end of a 12 cm rod whose ends are on the x- and y-axes.
Intercepts
Let the x- and y-intercepts of the rod ends be represented by 'a' and 'b', respectively. The fixed length of the rod tells us ...
a² +b² = 12²
according to the Pythagorean theorem.
Point P
The location of point P is 3/12 = 1/4 of the way from the x-intercept to the y-intercept. Its coordinates in terms of 'a' and 'b' are ...
P = 3/4(a, 0) +1/4(0, b) = (3a/4, b/4)
Equation of locus
If we define the point P as having coordinates (x, y), then we have ...
3a/4 = x ⇒ a = 4/3x
b/4 = y ⇒ b = 4y
Using these values in the above relation between 'a' and 'b', we have ...
(4/3x)² +(4y)² = 12²
We can divide by 12² to get the following equation of the ellipse that is the locus of P.
(x/9)² +(y/3)² = 1 . . . . . . useful domain/range: 0≤x≤9; 0≤y≤3.
