Find the condition that m and c satisfy if y = mx + c touches the circle x² + y² - 2ax = 0

To find the condition that \( m \) and \( c \) satisfy if the line \( y = mx + c \) touches the circle \( x^2 + y^2 - 2ax = 0 \), follow these steps:

### Step 1: Equation of the Circle
The given equation of the circle is:
\[
x^2 + y^2 - 2ax = 0
\]
This can be rewritten in the standard form of a circle:
\[
(x - a)^2 + y^2 = a^2
\]
This represents a circle with center \( (a, 0) \) and radius \( a \).

### Step 2: Condition for Tangency
For the line \( y = mx + c \) to touch the circle, the distance between the center of the circle and the line must be equal to the radius of the circle.

The distance \( d \) from the center \( (a, 0) \) to the line \( y = mx + c \) is given by the formula:
\[
d = \frac{|ma + c|}{\sqrt{1 + m^2}}
\]
Since the line touches the circle, this distance must be equal to the radius of the circle, which is \( a \). Therefore, we have the condition:
\[
\frac{|ma + c|}{\sqrt{1 + m^2}} = a
\]

### Step 3: Solve for Condition
To solve for the condition that \( m \) and \( c \) must satisfy, multiply both sides of the equation by \( \sqrt{1 + m^2} \):
\[
|ma + c| = a \sqrt{1 + m^2}
\]
This is the required condition for the line to be tangent to the circle.

Thus, the line \( y = mx + c \) touches the circle \( x^2 + y^2 - 2ax = 0 \) if \( m \) and \( c \) satisfy the condition:
\[
|ma + c| = a \sqrt{1 + m^2}
\]

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Would you like more details on any of these steps? Here are five related questions to explore further:

1. How do you derive the distance formula between a point and a line?
2. What happens if the line does not touch the circle but intersects it?
3. Can you explore the geometry behind tangents and circles in greater depth?
4. What would the condition be if the center of the circle were at a different point?
5. How does the equation change if the circle's radius is not \( a \) but some other value?

**Tip:** When working with tangency conditions, always equate the distance between the point and the line to the radius for clarity in deriving the condition.

Discover the condition for a line y = mx + c to touch the circle x² + y² - 2ax = 0. Learn the geometry of tangency, including the distance formula between a point and a line. Suitable for advanced high school students and anyone interested in circle-line tangency problems.

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