1. **Problem Statement:** Given a circle with center $P$, segment $EA$ is tangent to the circle at point $A$. We know $EA=24$ and chord $CD=14$. We need to find the length of segment $EC$. 2. **Relevant Theorem:** The tangent-secant theorem states that if a tangent from a point outside the circle and a secant from the same point intersect the circle, then the square of the tangent segment equals the product of the entire secant segment and its external part. Mathematically, if $EA$ is tangent and $ECD$ is a secant, then: $$EA^2 = EC \times ED$$ 3. **Identify segments:** We know $EA=24$, $CD=14$. Since $C$ and $D$ lie on the circle, $ED = EC + CD = EC + 14$. 4. **Apply the theorem:** $$24^2 = EC \times (EC + 14)$$ $$576 = EC^2 + 14EC$$ 5. **Rewrite as quadratic equation:** $$EC^2 + 14EC - 576 = 0$$ 6. **Solve quadratic equation using the quadratic formula:** $$EC = \frac{-14 \pm \sqrt{14^2 - 4 \times 1 \times (-576)}}{2}$$ $$= \frac{-14 \pm \sqrt{196 + 2304}}{2} = \frac{-14 \pm \sqrt{2500}}{2}$$ $$= \frac{-14 \pm 50}{2}$$ 7. **Calculate roots:** $$EC_1 = \frac{-14 + 50}{2} = \frac{36}{2} = 18$$ $$EC_2 = \frac{-14 - 50}{2} = \frac{-64}{2} = -32$$ Since length cannot be negative, $EC = 18$. **Final answer:** $$\boxed{18}$$