If a circle passing through (1, 2) and intersects $x^{2}+y^{2}=4$ orthogonally, then the locus of the centre of the circle is:

Question

Accepted Answer

To determine the locus of the center of a circle that passes through the point (1, 2) and intersects the circle given by $x^2 + y^2 = 4$ orthogonally, we need to use the concept of orthogonal circles.

### Explanation:

1. **Orthogonal Circles Condition:**
   Two circles are orthogonal if the sum of the products of their corresponding coefficients is zero. For a circle with center $(h, k)$ and radius $r$, the equation is $(x - h)^2 + (y - k)^2 = r^2$, which expands to:
   $$
   x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0
   $$

The given circle is $x^2 + y^2 = 4$, which can be rewritten as:
   $$
   x^2 + y^2 + 0x + 0y - 4 = 0
   $$

For orthogonality, the condition is:
   $$
   2h \cdot 0 + 2k \cdot 0 + (h^2 + k^2 - r^2) \cdot (-4) = 0
   $$
   Simplifying, we get:
   $$
   h^2 + k^2 - r^2 = 4
   $$

2. **Circle Passing Through (1, 2):**
   The circle also passes through the point (1, 2), so substituting these coordinates into the circle's equation:
   $$
   (1 - h)^2 + (2 - k)^2 = r^2
   $$
   Expanding this, we have:
   $$
   (1 - h)^2 + (2 - k)^2 = r^2 \Rightarrow 1 - 2h + h^2 + 4 - 4k + k^2 = r^2
   $$
   $$
   h^2 + k^2 - 2h - 4k + 5 = r^2
   $$

3. **Eliminating $r^2$:**
   From the orthogonality condition, $h^2 + k^2 - r^2 = 4$, we substitute $r^2$ from the equation above:
   $$
   h^2 + k^2 - (h^2 + k^2 - 2h - 4k + 5) = 4
   $$
   Simplifying, we get:
   $$
   2h + 4k - 5 = 4
   $$
   $$
   2h + 4k = 9
   $$

4. **Locus of the Center:**
   The equation $2h + 4k = 9$ represents the locus of the center of the circle. This is a linear equation in $h$ and $k$, which matches Option 3:
   $$
   2x + 4y - 9 = 0
   $$

### Why Other Options are Incorrect:

- **Option 1: $x^2 + y^2 - 3x - 8y + 1 = 0$**
  - This represents a circle, not a line, and does not satisfy the derived linear equation for the locus.

- **Option 2: $x^2 + y^2 - 2x - 6y - 7 = 0$**
  - Similar to Option 1, this is a circle equation, not a line, and does not match the derived condition.

- **Option 4: $2x + 4y - 1 = 0$**
  - This is a linear equation but does not satisfy the derived condition $2x + 4y = 9$.

Thus, Option 3 is the correct answer as it accurately represents the locus of the center of the circle that meets the given conditions.