CBSE 2023 · Region 3 · Set 1 · Q22 · 2 marks
If the angle between the lines $\displaystyle \frac{\mathrm{x}-5}{\alpha}=\frac{\mathrm{y}+2}{-5}=\frac{\mathrm{z}+\dfrac{24}{5}}{\beta}$ and $\displaystyle \frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{0}=\frac{\mathrm{z}}{1}$ is $\displaystyle \frac{\pi}{4}$, find the relation between $\displaystyle \alpha$ and $\displaystyle \beta$.
Marking-scheme solution
$$\cos \frac{\pi}{4}=\frac{|\alpha \cdot 1+0+\beta|}{\sqrt{\alpha^{2}+\beta^{2}+25} \sqrt{2}} \Rightarrow \frac{1}{\sqrt{2}}=\frac{|\alpha+\beta|}{\sqrt{\alpha^{2}+\beta^{2}+25} \sqrt{2}}
Squaring both sides, we get
\begin{aligned}
& \alpha^{2}+\beta^{2}+2 \alpha \beta=\alpha^{2}+\beta^{2}+25 \\
& \Rightarrow \alpha \beta=\frac{25}{2}
\end{aligned}
$$
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CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.