Example of translation in geometry with formula and coordinate transformation.

It looks like you've uploaded a document about geometry transformations, specifically focusing on translation. The document explains the concept, provides formulas, examples, and asks for solutions to problems.

Let's go through the content step by step and solve the problem that is partially shown — particularly the one involving the circle under translation.

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🔷 Problem:

> The image point of circle $ x^2 + y^2 = 25 $ by translation $ T = (-1, 3) $ is ...

We are asked to find the equation of the image of the circle after applying the translation $ T = (-1, 3) $.

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✔ Step-by-Step Solution:

#### 🔹 Step 1: Understand the Original Circle

The given equation is:
$$
x^2 + y^2 = 25
$$
This is a circle centered at the origin $ (0, 0) $ with radius $ r = \sqrt{25} = 5 $.

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#### 🔹 Step 2: Apply the Translation $ T = (-1, 3) $

A translation $ T = (a, b) = (-1, 3) $ moves every point $ (x, y) $ to:
$$
(x', y') = (x + a, y + b) = (x - 1, y + 3)
$$

So, to find the image of the circle, we need to express the original coordinates $ (x, y) $ in terms of the new (translated) coordinates $ (x', y') $.

Let:
- $ x' = x - 1 $ → $ x = x' + 1 $
- $ y' = y + 3 $ → $ y = y' - 3 $

Now substitute these into the original equation $ x^2 + y^2 = 25 $:

$$
(x)^2 + (y)^2 = 25 \\
(x' + 1)^2 + (y' - 3)^2 = 25
$$

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#### 🔹 Step 3: Write the Equation of the Image

Replace $ x' $ and $ y' $ with $ x $ and $ y $ (standard notation for final equation):

$$
(x + 1)^2 + (y - 3)^2 = 25
$$

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✔ Final Answer:

The image of the circle $ x^2 + y^2 = 25 $ under the translation $ T = (-1, 3) $ is:
$$
\boxed{(x + 1)^2 + (y - 3)^2 = 25}
$$

This represents a circle with center at $ (-1, 3) $ and radius 5.

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📌 Explanation:

- Translation shifts every point of a shape by the same vector.
- The center of the circle $ (0, 0) $ moves to $ (-1, 3) $.
- The radius remains unchanged.
- So, the new equation reflects the new center while keeping the same radius.

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Let me know if you'd like this explained visually or want help with any other part of the document!