A triangle ABC with area 5a^{2} is inscribed | vedclass

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(D) Let the coordinates of the endpoints of the focal chord $BC$ be $B(at_{1}^{2}, 2at_{1})$ and $C(at_{2}^{2}, 2at_{2})$.Since $BC$ is a focal chord,

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$25a$

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(D) Let the coordinates of the endpoints of the focal chord $BC$ be $B(at_{1}^{2}, 2at_{1})$ and $C(at_{2}^{2}, 2at_{2})$.Since $BC$ is a focal chord,$t_{1}t_{2} = -1$.The area of $\Delta ABC$ with vertices $A(0, 0)$,$B(at_{1}^{2}, 2at_{1})$,and $C(at_{2}^{2}, 2at_{2})$ is given by:$	ext{Area} = \frac{1}{2} |x_{A}(y_{B} - y_{C}) + x_{B}(y_{C} - y_{A}) + x_{C}(y_{A} - y_{B})|$$	ext{Area} = \frac{1}{2} |0 + at_{1}^{2}(2at_{2} - 0) + at_{2}^{2}(0 - 2at_{1})| = \frac{1}{2} |2a^{2}t_{1}^{2}t_{2} - 2a^{2}t_{1}t_{2}^{2}|$$	ext{Area} = a^{2} |t_{1}t_{2}(t_{1} - t_{2})| = a^{2} |-1(t_{1} - t_{2})| = a^{2} |t_{1} - t_{2}|$.Given that the area is $5a^{2}$,we have $a^{2} |t_{1} - t_{2}| = 5a^{2}$,which implies $|t_{1} - t_{2}| = 5$.The length of the focal chord is $L = a(t_{1} - t_{2})^{2}$.Since $t_{2} = -1/t_{1}$,we have $t_{1} - t_{2} = t_{1} + 1/t_{1}$.Thus,$(t_{1} - t_{2})^{2} = (t_{1} + 1/t_{1})^{2} = (t_{1} - 1/t_{1})^{2} + 4 = (t_{1} - t_{2})^{2} + 4 = 5^{2} + 4 = 29$.Wait,the length of the focal chord is $a(t_{1} - t_{2})^{2} = a(t_{1} + 1/t_{1})^{2}$.Using $|t_{1} - t_{2}| = 5$,we have $(t_{1} - t_{2})^{2} = 25$.Since $t_{1}t_{2} = -1$,$(t_{1} + t_{2})^{2} = (t_{1} - t_{2})^{2} + 4t_{1}t_{2} = 25 - 4 = 21$.The length of the focal chord $BC$ is $a(t_{1} - t_{2})\sqrt{(t_{1} + t_{2})^{2}} = a(5)\sqrt{21}$ is not correct.The length of the focal chord is $a(t_{1} - t_{2})^{2} = a(t_{1} + 1/t_{1})^{2} = a(t_{1}^{2} + 1/t_{1}^{2} + 2) = a((t_{1} - 1/t_{1})^{2} + 4) = a(25 + 4) = 29a$.Re-evaluating: The length of the focal chord is $a(t_{1} + 1/t_{1})^{2}$. Given $|t_{1} - t_{2}| = 5$,where $t_{2} = -1/t_{1}$,then $|t_{1} + 1/t_{1}| = 5$. Thus,length $= a(5)^{2} = 25a$.

$A$ triangle $ABC$ with area $5a^{2}$ is inscribed in the parabola $y^{2} = 4ax$,where the vertex $A$ is at the vertex of the parabola and $BC$ is a focal chord. Find the length of the focal chord.

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