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If \mathrm{A\left(t_1\right)} and \mathrm{B\left(t_2\right)} be the two points on the parabola \mathrm{y^2=4 a x} and \mathrm{A B} cuts the x-axis at \mathrm{C} such that\mathrm{ A B: A C=3: 1} and the chord \mathrm{ A B} is parallel to the line \mathrm{ y=2 x+3}, find the coordinates of \mathrm{ A} and \mathrm{ B}.

Option: 1

\mathrm{A \equiv(a,-2 a), B \equiv(4 a, 4 a)}


Option: 2

\mathrm{A \equiv\left(\frac{a}{4}, a\right), B \equiv(4 a,-4 a)}


Option: 3

\mathrm{A \equiv(a, 2 a), \quad B \equiv(4 a, 4 a)}


Option: 4

\mathrm{A \equiv\left(\frac{a}{4},-a\right), \quad B \equiv(4 a, 4 a)}


Answers (1)

best_answer

\mathrm{ \begin{aligned} & A B: A C=3: 1 \\ & \Rightarrow B C: A C=2: 1 \\ & \qquad \frac{4 \mathrm{at}_1+2 a t_2}{3}=0 \Rightarrow 2 t_1+t_2=0 \end{aligned} }
Hence

Since \mathrm{A B} is parallel to \mathrm{y=2 x+3}

\mathrm{ \begin{aligned} & \frac{2 a\left(t_1-t_2\right)}{a\left(t_1^2-t_2^2\right)}=2 \\ & \Rightarrow t_1+t_2=1 \\ & \text { Hence } t_1=-1, t_2=2 \\ & \Rightarrow A \equiv(a,-2 a), B \equiv(4 a, 4 a) . \end{aligned} }

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Ajit Kumar Dubey

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