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The problem you’ve provided discusses two circles and asks to find the equation of the tangent \( CBD \). Let's break it down based on the information given:

### Information:
1. **First circle**: Center \( A(-1, 4) \), passes through point \( B(3, 2) \).
2. **Second circle**: Center \( B(3, 2) \), passes through point \( A(-1, 4) \).
3. The **tangent** at \( B \) to the first circle intersects the second circle at points \( C \) and \( D \).

#### (a) To find the equation of the tangent \( CBD \):

We know that the tangent at point \( B \) is perpendicular to the radius at \( B \), so we need to:
1. Find the **slope of the radius** from \( A \) to \( B \). This is the slope of the line passing through \( A(-1, 4) \) and \( B(3, 2) \):
   \[
   m_{\text{AB}} = \frac{2 - 4}{3 - (-1)} = \frac{-2}{4} = -\frac{1}{2}
   \]
2. The slope of the **tangent** will be the negative reciprocal of the slope of \( AB \), because the tangent is perpendicular to the radius:
   \[
   m_{\text{tangent}} = 2
   \]
3. Now, using the point-slope form of a line equation \( y - y_1 = m(x - x_1) \), where the point is \( B(3, 2) \) and the slope is 2:
   \[
   y - 2 = 2(x - 3)
   \]
   Simplifying:
   \[
   y - 2 = 2x - 6
   \]
   \[
   y = 2x - 4
   \]
Thus, the equation of the tangent \( CBD \) is:
\[
y = 2x - 4
\]

Would you like a detailed breakdown of the next part or have any questions on this?

### Follow-up Questions:
1. How do we find the radius of a circle given two points?
2. What is the general equation of a circle?
3. How do we determine if a line is tangent to a circle geometrically?
4. What is the significance of the point of tangency in this context?
5. How can we calculate the coordinates of \( C \) and \( D \)?

### Tip:
Remember, the slope of a tangent to a circle is always the negative reciprocal of the slope of the radius at the point of tangency!

The diagram shows a circle with center A passing through point B. A second circle has center B and passes through A. The tangent at B to the first circle intersects the second circle at C and D. The coordinates of A are (-1, 4) and the coordinates of B are (3, 2). (a) Find the equation of the tangent CBD.

Learn how to find the equation of the tangent line to two intersecting circles. The problem involves calculating slopes, applying geometry principles, and using point-slope form. Ideal for high school students in grades 10-12.

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