Finding the Value of (a) for a Circle’s Diameter

Consider the circle equation x2 - 2xy2 - 4y - 4 0. If a line 2x - ya 0 is the diameter of this circle, what is the value of a?

Analysis Solution

To solve the problem, let's first transform the circle equation into a standard form. We start with:

x2 - 2xy2 - 4y - 4 0

First, we complete the square for both (x) and (y):

x2 - 2x(1)y2 - 4y - 4 0

x2 - 2xy2 - 4y - 4 0

x2 - 2x 1 - 1 - 2y2 - 4y - 4 0

(x - 1)2 - 1 - 2(y - 2)2 8 - 4 0

(x - 1)2 - 2(y - 2)2 3 0 ? (x - 1)2 - 2(y - 2)2 -3

(x - 1)2/3 - 2(y - 2)2/3 1

We can see that this is not in the correct standard form for a circle; the correct form for a circle should be:

(x - h)2 (y - k)2 r2

To proceed, let's rewrite the original equation in the correct form:

x2 - 2x - 2(xy2 2y 4) 0

(x - 1)2 (y - 2)2 9

Therefore, the center of the circle is at (1, 2).

Given the diameter is the line 2x - ya 0, the point (1, 2) must satisfy this equation. Plug x 1 and y 2 into the line equation:

2(1) - 2a 0

2 - 2a 0

2a 2

a 1

However, the problem specifies that it must be 2x - ya 0, which implies the simplified form is 2x - 2y 0. Plugging in these points:

2(1) - 2(2) 0

2 - 4 0

-2 0

This contradiction suggests it must be differently approached; the values simplify to a 0.

Therefore, the value of (a) is 0.