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प्रश्न
A straight rod of length 8 units slides with its ends A and B always on the x and y axes respectively. Find the locus of the midpoint of the line segment AB
उत्तर
Given A and B are the ends of the straight rod of length 8 unit on the x and y-axes.
Let A be (a, 0) and B(0, b).
Let M(h, k) be the midpoint of AB(h, k)
= (h, k) = `(("a" + 0)/2, (0 +"b")/2)`
= `("a"/2, "b"/2)`
h = `"a"/2`
k = `"b"/2`
⇒ a = 2h, b = 2k
In the right-angled ∆OAB
AB2 = OA2 + OB2
82 = a2 + b2
64 = (2h)2 + (2k)
64 = 4h2 + 4k2
h2 + k2 = `64/4` = 16
The locus of M (h, k) is obtained by replacing h by x and k by y
∴ The required locus is x2 + y2 = 16
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