Consider the lines L₁ : x − 1 = y − 2 = z and L₂ : x − 2 = y = z − 1. Let the feet of the perpendiculars from the point P(5, 1, −3) on the lines L₁ and L₂ be Q and R respectively. If the area of the triangle PQR is A, then 4A² is equal to:
Question: Consider the lines L 1 : x − 1 = y − 2 = z L_{1} : x - 1 = y - 2 = z and L 2 : x − 2 = y = z − 1 L_{2} : x - 2 = y = z - 1 Let the feet of the perpendiculars from the point P ( 5 , 1 , − 3 ) P(5,\,1,\,-3) on the lines L 1 L_{1} and L 2 L_{2} be Q Q and R R respectively. If the area of the triangle P Q R PQR is A A , then find 4 A 2 4A^{2} . Question Image Short Solution To find 4 A 2 4A^{2} , follow these steps: Write lines in parametric form L 1 : x = 1 + t , y = 2 + t , z = t L 2 : x = 2 + s , y = s , z = 1 + s Find the foot of the perpendicular from P P to each line Use the formula for foot of perpendicular on a line: Q = A + ( P − A ) ⋅ d ∣ d ∣ 2 d Q = A + \dfrac{(P-A)\cdot d}{|d|^{2}} d where A A is a point on the line and d d its direction vector. Get coordinates of Q Q and R R by solving the dot product condition. Find the area of triangle P Q R P...