Question:
The two lines x = ay + b, z = cy + d; and x = a’y + b’, z = c’y + d’ are perpendicular to each other, if:
दो रेखाएँ x = ay + b, z = cy + d; और x = a’y + b’, z = c’y + d’ एक-दूसरे के लंबवत हैं, यदि:
Options:
- \(\dfrac{a}{a’} + \dfrac{c}{c’} = 1\)
- \(\dfrac{a}{a’} + \dfrac{c}{c’} = -1\)
- aa’ + cc’ = 1
- aa’ + cc’ = −1
- \(\dfrac{a}{a’} + \dfrac{c}{c’} = 1\)
- \(\dfrac{a}{a’} + \dfrac{c}{c’} = -1\)
- aa’ + cc’ = 1
- aa’ + cc’ = −1
Correct Answer: d)
Explanation:
Direction ratios of the lines are \((a, 1, c)\) and \((a’, 1, c’)\).
Perpendicular condition: \(aa’ + (1)(1) + cc’ = 0 \implies aa’ + cc’ = -1\).
रेखाओं के दिक्-अनुपात \((a, 1, c)\) और \((a’, 1, c’)\) हैं। लंबवत होने पर: \(aa’ + 1 + cc’ = 0 \implies aa’ + cc’ = -1\)।