Let Q and R be the feet of perpendiculars from the point P a , a , a on the lines x = y , z = 1 and x = − y , z = − 1 respectively. If ∠ Q P R is a right angle, then 12 a 2 is equal to ________

Step-by-step solution

Given, Equation of line x = y , z = 1 x 1 = y 1 = z − 1 0 = r Now, let any point on the above line be Q r , r , 1 Similarly, for line x = − y , z = − 1 we get, x 1 = y − 1 = z + 1 0 = k So, any point on the above line will be R k , − k , − 1 Now, the direction ratio of P Q → = a − r i ^ + a − r j ^ + a − 1 k ^ Now, finding foot of perpendicular from P on line x 1 = y 1 = z − 1 0 using perpendicular condition we get, a - r + a − r + 0 = 0 ⇒ 2 a = 2 r ⇒ a = r Hence, point Q a , a , 1 And similarly, the direction rati