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प्रश्न
The equation of the circle concentric with x2 + y2 − 3x + 4y − c = 0 and passing through (−1, −2) is
विकल्प
x2 + y2 − 3x + 4y − 1 = 0
x2 + y2 − 3x + 4y = 0
x2 + y2 − 3x + 4y + 2 = 0
none of these
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उत्तर
x2 + y2 − 3x + 4y = 0
The centre of the circle x2 + y2 − 3x + 4y − c = 0 is \[\left( \frac{3}{2}, - 2 \right)\].
Therefore, the centre of the required circle is \[\left( \frac{3}{2}, - 2 \right)\].
The equation of the circle is \[\left( x - \frac{3}{2} \right)^2 + \left( y + 2 \right)^2 = a^2\] ...(1)
Also, circle (1) passes through (−1, −2).
\[\therefore \left( - 1 - \frac{3}{2} \right)^2 + \left( - 2 + 2 \right)^2 = a^2\]
⇒ \[a = \frac{5}{2}\]
Substituting the value of a in equation (1):
\[\left( x - \frac{3}{2} \right)^2 + \left( y + 2 \right)^2 = \left( \frac{5}{2} \right)^2 \]
\[ \Rightarrow \frac{\left( 2x - 3 \right)^2}{4} + \left( y + 2 \right)^2 = \frac{25}{4}\]
\[ \Rightarrow \left( 2x - 3 \right)^2 + 4 \left( y + 2 \right)^2 = 25\]
\[ \Rightarrow x^2 + y^2 - 3x + 4y = 0\]
Hence, the required equation of the circle is \[x^2 + y^2 - 3x + 4y = 0\].
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