If the pole of the line with respect to the circle `x^(2)+y^(2)=c^(2)` lies on the circle `x^(2)+y^(2)=9c^(2)` then the line is a tangent to the circle with centre origin is

Options

A. `x^(2)+y^(2)=9c^(2)`
B. `9x^(2)+9y^(2)=c^(2)`
C. `3x^(2)+3y^(2)=c^(2)`
D. `81x^(2)+81y^(2)=c^(2)`

Answer

To solve the problem step-by-step, we will analyze the given information and derive the required equation for the line that is a tangent to the circle with center at the origin.

### Step 1: Understand the given circles
We have two circles:
1. Circle 1: \( x^2 + y^2 = c^2 \)
2. Circle 2: \( x^2 + y^2 = 9c^2 \)

### Step 2: Identify the pole of the line
The pole of a line with respect to a circle is a point that is related to the line's position concerning the circle. The equation of the line can be represented in the form:
\[ xx_1 + yy_1 = c^2 \]
where \( (x_1, y_1) \) is the pole of the line.

### Step 3: Condition for the pole
According to the problem, the pole \( (x_1, y_1) \) lies on Circle 2. Therefore, we have:
\[ x_1^2 + y_1^2 = 9c^2 \]

### Step 4: Relate the two circles
From the equations of the circles, we know:
- For Circle 1, the radius is \( c \).
- For Circle 2, the radius is \( 3c \).

### Step 5: Tangent condition
For the line to be tangent to Circle 1, the distance from the center of Circle 1 (which is the origin) to the line must equal the radius of Circle 1.

The distance \( d \) from the center (0, 0) to the line \( ax + by + c = 0 \) is given by:
\[ d = \frac{|c|}{\sqrt{a^2 + b^2}} \]

### Step 6: Set up the tangent condition
For the line to be tangent to Circle 1, we set:
\[ d = c \]
Thus, we have:
\[ \frac{|c|}{\sqrt{a^2 + b^2}} = c \]

### Step 7: Simplify the equation
By simplifying the above equation, we can cancel \( c \) (assuming \( c \neq 0 \)):
\[ |c| = c \sqrt{a^2 + b^2} \]
Squaring both sides gives:
\[ c^2 = c^2 (a^2 + b^2) \]

### Step 8: Analyze the result
This implies:
\[ 1 = a^2 + b^2 \]
This means that the coefficients of the line must satisfy this equation.

### Step 9: Conclusion
Thus, the line that is tangent to the circle \( x^2 + y^2 = c^2 \) must have coefficients \( a \) and \( b \) such that \( a^2 + b^2 = 1 \).

### Final Answer
The line is a tangent to the circle with center at the origin when the pole of the line lies on the circle \( x^2 + y^2 = 9c^2 \).