Question

Please help me solve this question in a different way.

I know one way of doing this question that is by using sin theta and cos theta. But I want to learn a new way to do this same question.

Answer 1

(x/9)² +(y/3)² = 1 . . . P = (x, y)

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.