The distance between the two points A and A' which lie on y=2 such that both the line segm…

Question

The distance between the two points $A$ and $A'$ which lie on $y=2$ such that both the line segments $AB$ and $A'B$ (where $B$ is the point $(2,3)$) subtend angle $\dfrac{\pi}4$ at the origin, is equal to:

Accepted Answer

Let $A(\alpha, 2) \quad$ Given $B(2,3)$

$$
\begin{aligned}
& m_{O A}=\frac{2}{\alpha} \quad&\quad m_{O B}=\frac{3}{2} \\
& \tan \frac{\pi}{4}=\left|\frac{\frac{2}{\alpha}-\frac{3}{2}}{1+\frac{2}{\alpha} \cdot \frac{3}{2}}\right| \Rightarrow \frac{4-3 \alpha}{2 \alpha+6}=\pm 1 \\
& 4-3 \alpha=2 \alpha+6 \quad & 4-3 \alpha=-2 \alpha-6 \\
& \alpha=\frac{-2}{5} & \alpha=10 \\
& A\left(-\frac{2}{5}, 2\right) & A^{\prime}(10,2) \text { and } B(2,3) \\
& A A^{\prime}=10+\frac{2}{5}=\frac{52}{5}
\end{aligned}
$$

Solutions · 1