The locus of a point which moves in a plane so that the sum of the squares of its distances from the line ax + by + c=0 and bx -ay + d=0 is `r ^(2),` is a circle of radius

Options

A. r
B. `r sqrt (a ^(2) + b ^(2))`
C. `r sqrt(ab)`
D. none of these

Answer

To solve the problem, we need to find the radius of the circle that represents the locus of a point in a plane such that the sum of the squares of its distances from the two given lines is equal to \( r^2 \).

### Step-by-Step Solution:

1. **Define the Point and Lines**:
Let the point be \( (h, k) \). The two lines are given by:
\[ L_1: ax + by + c = 0 \]
\[ L_2: bx - ay + d = 0 \]

2. **Calculate the Distances**:
The perpendicular distance \( d_1 \) from the point \( (h, k) \) to the line \( L_1 \) is given by:
\[ d_1 = \frac{|ah + bk + c|}{\sqrt{a^2 + b^2}} \]
The perpendicular distance \( d_2 \) from the point \( (h, k) \) to the line \( L_2 \) is given by:
\[ d_2 = \frac{|bh - ak + d|}{\sqrt{b^2 + a^2}} \]

3. **Set Up the Equation**:
According to the problem, the sum of the squares of these distances equals \( r^2 \):
\[ d_1^2 + d_2^2 = r^2 \]
Substituting the distances, we have:
\[ \left(\frac{ah + bk + c}{\sqrt{a^2 + b^2}}\right)^2 + \left(\frac{bh - ak + d}{\sqrt{b^2 + a^2}}\right)^2 = r^2 \]

4. **Simplify the Equation**:
Squaring both distances gives:
\[ \frac{(ah + bk + c)^2}{a^2 + b^2} + \frac{(bh - ak + d)^2}{b^2 + a^2} = r^2 \]
Multiply through by \( (a^2 + b^2) \) to eliminate the denominators:
\[ (ah + bk + c)^2 + (bh - ak + d)^2 = r^2(a^2 + b^2) \]

5. **Expand the Squares**:
Expanding both sides, we get:
\[ (a^2h^2 + 2abhk + b^2k^2 + 2ack + c^2) + (b^2h^2 - 2abhk + a^2k^2 + 2bdh + d^2) = r^2(a^2 + b^2) \]

6. **Combine Like Terms**:
Combine the terms on the left side:
\[ (a^2 + b^2)h^2 + (a^2 + b^2)k^2 + 2(abhk) + (c^2 + d^2 + 2bdh + 2ack) = r^2(a^2 + b^2) \]

7. **Rearranging**:
Rearranging gives:
\[ (a^2 + b^2)(h^2 + k^2) + 2(abhk + ack + bdh) + (c^2 + d^2 - r^2) = 0 \]

8. **Identify Circle Parameters**:
This equation represents a circle in the form:
\[ (h - h_0)^2 + (k - k_0)^2 = R^2 \]
where \( h_0 \) and \( k_0 \) are the center coordinates, and \( R \) is the radius.

9. **Finding the Radius**:
The radius \( R \) can be determined from the constant term and the coefficients of \( h \) and \( k \):
\[ R = \sqrt{g^2 + f^2 - c} \]
where \( g \) and \( f \) are derived from the coefficients of \( h \) and \( k \), and \( c \) is the constant term.

### Final Result:
The radius of the circle is given by:
\[ R = \frac{r}{\sqrt{a^2 + b^2}} \]