Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade None
Question:
If one vertex of an equilateral triangle of side '$a$' is at $(1,0)$, also another one lies on the line $\sqrt{3}x - y - \sqrt{3} = 0$, then co-ordinates of the third vertex may be
left(1 - \frac{a}{2}, \frac{a\sqrt{3}}{2}\right)
left(\frac{a}{2} + 1, -\frac{a\sqrt{3}}{2}\right)
(a + 1,0)
(a - 1,0)
Step-by-Step Solution
Key Concept: Equilateral triangles have all angles $60°$ and symmetric properties that relate coordinates through height $\frac{a\sqrt{3}}{2}$.
An equilateral triangle is constructed with vertices at $\left(1-\frac{a}{2}, \frac{a\sqrt{3}}{2}\right)$, $\left(1+\frac{a}{2}, \frac{a\sqrt{3}}{2}\right)$, and $\left(1+\frac{a}{2}, -\frac{a\sqrt{3}}{2}\right)$, with base $B(1,0)$ and side length $a$. The configuration uses $60°$ angles and symmetric placement about the line through $B$.
Correct Answer: 1,2,3