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प्रश्न
If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.
If a point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), then find the value of p.
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उत्तर १
The given points are A(0, 2), B(3, p) and C(p, 5).
It is given that A is equidistant from B and C.
∴ AB = AC
⇒ AB2 = AC2
⇒ (3 − 0)2 + (p − 2)2 = (p − 0)2 + (5 − 2)2
⇒ 9 + p2 + 4 − 4p = p2 + 9
⇒ 4 − 4p = 0
⇒ 4p = 4
⇒ p = 1
Thus, the value of p is 1.
Length of AB `=sqrt((3-0)^2+(1-2)^2)=sqrt(3^2+(-1)^2)=sqrt(9+1)=sqrt(10) units`
उत्तर २
It is given that A(0, 2) is equidistant from the points B(3, p) and C(p, 5).
∴ AB = AC
\[\Rightarrow \sqrt{\left( 3 - 0 \right)^2 + \left( p - 2 \right)^2} = \sqrt{\left( p - 0 \right)^2 + \left( 5 - 2 \right)^2}\] (Distance formula)
Squaring on both sides, we get
\[9 + p^2 - 4p + 4 = p^2 + 9\]
\[ \Rightarrow - 4p + 4 = 0\]
\[ \Rightarrow p = 1\]
Thus, the value of p is 1.
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