Given four parallel lines L 1 , L 2 , L 3 and L 4 . Let the distances between them be d 12, d 23 , d 34 respectively. Let P be a point such of whose distances from four lines is k(d 12 < d 23 < d .34 ). Then the locus of the point P
Column I | Column II |
(i) If k = d12 + 2d23 + d34 | [A] Not possible |
(ii) If k = d12 + 2d23 + d34 + 2α where 0 < α < d12 | [B] Entire region between the lines L2 and L3 |
(iii) If k = d12 + 2d23 + d34 + 2α where 0 < α < d34 | [C] Entire region between the lines L1 and L2 |
(iv) If k < d12 + 2d23 + d34 | [D] Entire region between the lines L1 and L2 and between L3 and L4. |
Text Solution
Verified by Experts(i) [B]; (ii) [C]; (iii) [D]; (iv) [A]
Ans.
(i) [B]
(ii) [C]
(iii) [D]
(iv) [A]
Sol. (i) For any point P lying between the lines L 2 and L 3 the sum of the distances of P from L 1 , L 2 , L 3 and L 4 is d 12 + 2d 23 + d 34
Clearly, there is no point possible with
k < d 12 + 2d 23 + d 34
If the point P lies between the lines L 1 and L 2 , then the sum of the distances of P from the four lines is
d 12 + 2d 23 + d 34 + 2 α where 0 < α < d 12
(ii) For any point P lying between the lines L 2 and L 3 the sum of the distances of P from L 1 , L 2 , L 3 and L 4 is d 12 + 2d 23 + d 34
Clearly, there is no point possible with k < d 12 + 2d 23 + d 34
If the point P lies between the lines L 1
and L 2 , then the sum of the distances of P from the four lines isd 12 + 2d 23 + d 34 + 2 α where 0 < α < d 12
(iii) For any point P lying between the lines L 2 and L 3 the sum of the distances of P from L 1 , L 2 , L 3 and L 4 is d 12 + 2d 23 + d 34
Clearly, there is no point possible with k < d 12 + 2d 23 + d 34
If the point P lies between the lines L 1 and L 2 , then the sum of the distances of P from the four lines is
d 12 + 2d 23 + d 34 + 2 α where 0 < α < d 12
(iv) For any point P lying between the lines L 2 and L 3 the sum of the distances of P from L 1 , L 2 , L 3 and L 4 is d 12 + 2d 23 + d 34
Clearly, there is no point possible with k < d 12 + 2d 23 + d 34
If the point P lies between the lines L 1 and L 2 , then the sum of the distances of P from the four lines is
d 12 + 2d 23 + d 34 + 2 α where 0 < α < d 12
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