Two ants Amar and Akbar are standing at the co-ordinates (3,7) and (9,1) respectively of a coordinate plane. Amar wants to visit the axis and then meet Akbar. What is the minimum distance he has to crawl to do this?
Answers
Answer:
10
Step-by-step explanation:
Two ants Amar and Akbar are standing at the co-ordinates (3,7) and (9,1) respectively of a coordinate plane. Amar wants to visit the axis and then meet Akbar. What is the minimum distance he has to crawl to do this
As ant touches Axis first
Let say Point Ant touches axes = (x , 0)
Distance Covered = √((x - 3)² + 7²) + √((x - 9)² + 1²)
= √(x² - 6x + 58) + √(x² -18x + 82)
Differentiating
= (1/2) (2x - 6)/√(x² - 6x + 58) + (1/2)(2x - 18)/√(x² -18x + 82)
= (x-3)/√(x² - 6x + 58) + (x - 9)/√(x² -18x + 82)
Putting = 0
=> (x-3)/√(x² - 6x + 58) = (9 - x)/√(x² -18x + 82)
Squaring Both sides
=> (x-3)²(x² - 18x + 82) = (9-x)²(x² - 6x + 58)
=> (x² -6x + 9)(x² - 18x + 82) = (x² - 18x + 81)(x² - 6x + 58)
=> (x² - 18x + 81)(x² -6x + 9) + (x² -6x + 9) = (x² - 18x + 81)(x² - 6x + 58)
=> (x² - 18x + 81)(x² -6x + 9 - (x² - 6x + 58)) + (x² -6x + 9) = 0
=> -49(x² - 18x + 81) + (x² -6x + 9) = 0
=> -48x² +876x -3960 = 0
=> 48x² -876x + 3960 = 0
=> 4x² - 73x + 330 = 0
=> 4x² - 40x - 33x + 330 = 0
=> 4x(x - 10) -33(x - 10) = 0
=> x = 10 or x = 33/4
x = 8.25
8.25 lies between 3 & 9
Distance covered by putting x = 8.25 in √(x² - 6x + 58) + √(x² -18x + 82)
= 10