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The equation of circle [tex]\( A \)[/tex] is [tex]\( x^2 + y^2 + Cx + Dy + E = 0 \)[/tex]. If the circle is moved horizontally to the left of the [tex]\( y \)[/tex]-axis without changing the radius, how are the coefficients [tex]\( C \)[/tex] and [tex]\( D \)[/tex] affected?

Sagot :

GinnyAnswer
Let's analyze the given situation step-by-step.

### Original Circle Equation
The original equation of the circle is:
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0. \][/tex]

### Moving the Circle Horizontally to the Left
When we move a circle horizontally to the left by a distance [tex]\(a\)[/tex], only the x-coordinate of the center of the circle changes, which impacts the [tex]\(C\)[/tex] coefficient.

### New Circle Equation After Movement
Let's apply the horizontal shift to the circle's equation. Let the circle be shifted to the left by a distance [tex]\(a\)[/tex]. The new position of any point [tex]\((x, y)\)[/tex] on the circle will be replaced by [tex]\((x + a, y)\)[/tex].

### Substitute [tex]\((x + a)\)[/tex] for [tex]\(x\)[/tex]
We'll substitute [tex]\(x + a\)[/tex] for [tex]\(x\)[/tex] in the original equation. The new equation is:
[tex]\[ (x + a)^2 + y^2 + C(x + a) + Dy + E = 0. \][/tex]

### Expand and Simplify the New Equation:
Expanding this equation, we get:
[tex]\[ (x + a)^2 = x^2 + 2ax + a^2. \][/tex]

So,
[tex]\[ x^2 + 2ax + a^2 + y^2 + C(x + a) + Dy + E = 0. \][/tex]
[tex]\[ x^2 + 2ax + a^2 + y^2 + Cx + Ca + Dy + E = 0. \][/tex]

### Group the Terms:
Now, let's group the terms involving [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
[tex]\[ x^2 + y^2 + (C+2a)x + Dy + (a^2 + Ca + E) = 0. \][/tex]

### Compare with Original Equation:
Let's compare the new equation:
[tex]\[ x^2 + y^2 + (C + 2a)x + Dy + (a^2 + Ca + E) = 0 \][/tex]
with the original equation:
[tex]\[ x^2 + y^2 + Cx + Dy + E = 0. \][/tex]

### Effect on Coefficients [tex]\(C\)[/tex] and [tex]\(D\)[/tex]:
1. Coefficient [tex]\(C\)[/tex]:
- The new coefficient of [tex]\(x\)[/tex] is [tex]\(C + 2a\)[/tex].
- Thus, [tex]\(C\)[/tex] changes to [tex]\(C + 2a\)[/tex].

2. Coefficient [tex]\(D\)[/tex]:
- The coefficient of [tex]\(y\)[/tex] remains [tex]\(D\)[/tex], as moving horizontally doesn't affect the [tex]\(y\)[/tex]-value.

In summary, when the circle is moved horizontally to the left by a distance [tex]\(a\)[/tex]:
- The coefficient [tex]\(C\)[/tex] increases by twice the distance shifted, i.e., it becomes [tex]\(C + 2a\)[/tex].
- The coefficient [tex]\(D\)[/tex] remains unchanged.

Therefore, the answer is:
When the circle is moved horizontally to the left, the coefficient [tex]\(C\)[/tex] increases by twice the distance shifted, and the coefficient [tex]\(D\)[/tex] remains unchanged.
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