If angle between the lines \( \displaystyle\frac{x-1}{\alpha}=\displaystyle\frac{y+2}{-5}=\displaystyle\frac{z+\frac{7}{3}}{\beta} \) and \( x=z,y=0 \) is \( \displaystyle\frac{\pi}{4} \), then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 2\,\alpha\,\beta=25 \)
The second line x = z, y = 0 has direction ratios (1, 0, 1). The first has direction ratios \( (\alpha,-5,\beta) \). Using
\( \displaystyle \cos\theta=\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}} \)
with \( \theta=\frac{\pi}{4} \):
\( \displaystyle \frac{\alpha+\beta}{\sqrt{\alpha^2+25+\beta^2}\cdot\sqrt{2}}=\frac{1}{\sqrt{2}} \)
\( \displaystyle (\alpha+\beta)^2=\alpha^2+25+\beta^2 \)
\( \displaystyle 2\alpha\beta=25 \)