Find the shortest distance between the lines whether they intersect or not x-5/4=y-7/- 5=z+3/-5
Answers
Step-by-step explanation:
Given lines are
(x – 3)/3 = (y – 8)/–1 = (z– 3)/1 = r1 (say) ……(1)
(x + 3)/–3 = (y +7)/2 = (z – 6)/4 = r2 (say) ……(2)
Any point on line (1) is of the form P (3r1 + 3, 8 – r1, r1 + 3)
and on line (2) is of the form Q (–3 – 3r2, 2r2 – 7, 4r2 + 6).
If PQ is line of shortest distance, then direction ratios of PQ
= (3r1 + 3) – (–3 – 3r2), (8 – r1) – (2r2 – 7), (r1+ 3) – (4r2 + 6)
i.e. 3r1 + 3r2 + 6, –r1 – 2r2 + 15, r1 – 4r2 – 3
As PQ is perpendicular to lines (1) and (2)
∴ 3(3r1 + 3r2 + 6) – 1(–r1 – 2r2 + 15) + 1(r1 – 4r2 - 3) = 0
⇒11r1 + 7r2 = 0 ……(3)
and –3(3r1 + 3r2 + 6) + 2(–r1 – 2r2 + 15) + 4(r1 – 4r2 - 3) = 0
i.e. 7r1 + 11r2 = 0 ……(4)
On solving equations (3) and (4), we get r1 = r2= 0.
So, point P = (3, 8, 3) and Q = (–3, –7, 6).
∴ Length of shortest distance PQ = √{(–3–3)2 + (–7–8)2 + (6–3)2} = 3√30
Direction ratios of shortest distance line are 2, 5, –1.
∴ Equation of shortest distance line is
x–3/2 = y–8/5 = z–3/–1.