17. Intense atmosphere between the countries, it is important to remain vigilant on our borders. And our solliers are
ever ready for that. A soldier is sitting at the place representing the point (3.2) and notices on enemy jet. He notices
the path and calculates it to be a parabola symmetrical to the yaxis and vertex of parabola is at point (0, 2) with
path represented by y = x + a. For the above situation answer the following:
(1) the value of a in path is
(a) 3
(c) 2
(ir) The distance of jet from any point on path, at a given time, from soldier is
(a) (-3) + (y + 2) (b) Vix - 3)2 + (x + 2) (c) 768-3)*+-2) (d) V-2) + (y - 3)
(iii) The expression for distance in terms of only is
(a) +x+1
(h) Vir-3)* +6+4)
(d) - 2)² + (x - 1)
(w) The value of such that the distance between jet and soldier is least is
1
1
6
() Minimum distance is
(@) V5 units
(b) 5 units
(©)
units
(d)
units
b
Answers
Answer:
i) (c) 2
ii) (c) root (x-3)^2 - (x-2)^2
iii) (c) root (x-3)^2 + x^4
iv) (a) x=1
v) (a) root 5
Given : A soldier is sitting at the place representing the point (3.2) and notices on enemy jet to be a parabola symmetrical to the y axis and vertex of parabola is at point (0, 2) with path represented by y = x² + a
To find : value of a in path
The distance of jet from any point on path, at a given time, from soldier
The expression for distance in terms of only x
The value of x such that the distance between jet and soldier is least
Minimum distance
Solution:
y = x² + a
Vertex is ( 0 , 2)
=> 2 = 0² + a
=> a = 2
Value of a is 2
hence y = x² + 2
The distance of jet from any point on path, at a given time, from soldier ( 3 , 2) is
√{( y - 2)² + ( x - 3)²}
= √{( x² + 2 - 2)² + ( x - 3)²}
= √{( x² )² + ( x - 3)²}
= √{x⁴ + ( x - 3)²}
distance between jet and soldier is least
=> √{x⁴ + ( x - 3)²} is least
=> x⁴ + ( x - 3)² is least
f(x) = x⁴ + ( x - 3)²
f'(x) = 4x³ + 2(x - 3)
4x³ + 2x - 6 = 0
2x³ + x - 3 = 0
=> (x - 1)(2x² + 2x + 3) = 0
2x² + 2x + 3 gives imaginary roots
Hence x = 1
f''(x) = 12x² + 2 > 0 Hence distance is least
Least distance = √{x⁴ + ( x - 3)²}
= √{1⁴ + ( 1 - 3)²}
= √5
value of a in path = 2
The distance of jet from any point on path, at a given time, from soldier = √{( y - 2)² + ( x - 3)²}
The expression for distance in terms of only x is √{x⁴ + ( x - 3)²}
The value of x such that the distance between jet and soldier is least x = 1
Minimum distance √5
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