# Parabolas within parabolas

Geometry Level 5

$$P_{o}$$ is the parabola $$y^{2}=4x$$ with vertex $$K(0,0)$$, $$A$$ and $$B$$ are points on $$P_{o}$$ where tangents are at right angles. Let $$C$$ be the centroid of $$\Delta ABK$$. The locus of $$C$$ is another parabola $$P_{1}$$. Now the process is repeated with $$P_{1}$$ then $$P_{2},P_{3}....$$ etc. Then the length of latus rectum of $$P_{10}$$ can be expressed as $$\frac{a}{b}$$ where $$a,b$$ are co prime natural numbers.

Find $$a+b$$ .

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