Square Radius of Circle Intersecting Axes Three Points

To find the square of the radius of the circle $C$, we analyze the geometric conditions given in the problem:

1. Centre and Origin Intersection: Let the centre of the circle $C$ be $(h, k)$, where $h > 0$ and $k > 0$ since it lies in the first quadrant. A circle intersecting two coordinate axes can have at most $2 + 2 = 4$ intersection points. For the circle to intersect the coordinate axes at exactly three points, one of the intersection points must be shared by both axes, which means the circle must pass through the origin $(0,0)$.

2. Equal Intercepts: The points of intersection with the $x$-axis are $(0,0)$ and $(2h, 0)$, giving an $x$-intercept length of $2h$. The points of intersection with the $y$-axis are $(0,0)$ and $(0, 2k)$, giving a $y$-intercept length of $2k$. Since the circle cuts off equal intercepts from the axes, we have: $2h = 2k \implies h = k$

3. Radius and Equation of the Circle: Since the circle passes through the origin $(0,0)$ and has centre $(h, h)$, the square of its radius $r$ is: $r^2 = h^2 + h^2 = 2h^2$

   The equation of the circle $C$ is: $(x - h)^2 + (y - h)^2 = 2h^2 \implies x^2 + y^2 - 2hx - 2hy = 0$

4. Chord Length Condition: The perpendicular distance $d$ from the centre $(h, h)$ to the line $x + y - 1 = 0$ is: $d = \frac{|h + h - 1|}{\sqrt{1^2 + 1^2}} = \frac{|2h - 1|}{\sqrt{2}}$

   The length of the chord cut by the line on the circle is given as $\sqrt{14}$: $\text{Length of chord} = 2\sqrt{r^2 - d^2} = \sqrt{14}$

   Squaring both sides gives: $4(r^2 - d^2) = 14 \implies r^2 - d^2 = \frac{7}{2}$

   Substituting $r^2 = 2h^2$ and $d^2 = \frac{(2h - 1)^2}{2}$: $2h^2 - \frac{(2h - 1)^2}{2} = \frac{7}{2}$

   Multiply the entire equation by $2$: $4h^2 - (2h - 1)^2 = 7$ $4h^2 - (4h^2 - 4h + 1) = 7$ $4h - 1 = 7 \implies 4h = 8 \implies h = 2$

5. Square of the Radius: Substituting $h = 2$ into the expression for $r^2$: $r^2 = 2h^2 = 2(2)^2 = 8$

Thus, the square of the radius of $C$ is $8$.