To find the intersection points of the two inequalities in the system, let's first interpret and analyze each inequality:

1. $x^2 + y^2 < 25$: This represents the interior of a circle centered at the origin with radius $5$.
2. $3x^2 - y^2 > 12$: This represents the region outside of a hyperbola centered at the origin with specific constraints on $x$ and $y$.

The question asks us to find the points of intersection between these two regions, specifically in the quadrant where $x > 0$ and $y < 0$.

Step-by-Step Solution:

1. Convert the inequalities to equalities to find the boundary curves:

   • For the circle: $x^2 + y^2 = 25$
   • For the hyperbola: $3x^2 - y^2 = 12$

2. Substitute to find intersection points:

   • From $x^2 + y^2 = 25$, solve for $y^2$: $y^2 = 25 - x^2$
   • Substitute $y^2 = 25 - x^2$ into $3x^2 - y^2 = 12$: $3x^2 - (25 - x^2) = 12$ $4x^2 - 25 = 12$ $4x^2 = 37$ $x^2 = \frac{37}{4}$ $x = \pm \frac{\sqrt{37}}{2}$
   • Since we need $x > 0$, we choose $x = \frac{\sqrt{37}}{2}$.

3. Find $y$ corresponding to $x = \frac{\sqrt{37}}{2}$:

   • Substitute $x = \frac{\sqrt{37}}{2}$ into $y^2 = 25 - x^2$: $y^2 = 25 - \frac{37}{4} = \frac{100 - 37}{4} = \frac{63}{4}$ $y = \pm \frac{\sqrt{63}}{2} = \pm \frac{3\sqrt{7}}{2}$
   • Since $y < 0$, we choose $y = -\frac{3\sqrt{7}}{2}$.

4. Intersection Point:

   • The intersection point in the quadrant where $x > 0$ and $y < 0$ is: $\left( \frac{\sqrt{37}}{2}, -\frac{3\sqrt{7}}{2} \right)$

Conclusion:

The correct answer is: $\left( \frac{\sqrt{37}}{2}, -\frac{3\sqrt{7}}{2} \right)$

Answer Choice:

The correct answer is the fourth option.

Would you like further details on any step?

Follow-Up Questions:

1. How do we determine the intersection of two inequalities geometrically?
2. What is the significance of choosing specific quadrants in these problems?
3. How can we visualize a circle and a hyperbola intersecting?
4. What methods can we use to solve similar inequalities involving different conic sections?
5. How can we apply these concepts to find feasible regions in optimization problems?

Tip:

When solving systems of inequalities involving conic sections, converting inequalities to equalities and solving for boundary points often reveals the intersection boundaries.