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प्रश्न
Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).
उत्तर
Let (h, k) be the centre of a circle with radius a.
Thus, its equation will be \[\left( x - h \right)^2 + \left( y - k \right)^2 = a^2\]
Given:
h = 1, k = 2
∴ Equation of the circle = \[\left( x - 1 \right)^2 + \left( y - 2 \right)^2 = a^2\]
Also, equation (1) passes through (4, 6).
∴\[\left( 4 - 1 \right)^2 + \left( 6 - 2 \right)^2 = a^2\]
\[\Rightarrow 9 + 16 = a^2 \]
\[ \Rightarrow a = 5 \left( \because a > 0 \right)\]
Substituting the value of a in equation (1):
\[ \Rightarrow x^2 - 2x + y^2 - 4y = 20\]
\[ \Rightarrow x^2 + y^2 - 2x - 4y - 20 = 0\]
Thus, the required equation of the circle is
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