Definition of a circle

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Question

The equation of a circle passing through $(1, 1)$ and points of intersection of $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is

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Solution

The equation of the required circle is
$x^{2} + y^{2} + 13 x - 3 y + λ (x^{2} + y^{2} + 2 x - \frac{7}{2} y - \frac{25}{2}) = 0$
This passes through (1, 1).
$∴ 12 + λ (- 12) = 0 \Rightarrow λ = 1 .$
Hence, the required circle is
$\begin{array}{l} x^{2} + y^{2} + 13 x - 3 y + x^{2} + y^{2} + 2 x - \frac{7}{2} y - \frac{25}{2} = 0 \\ \Rightarrow 4 x^{2} + 4 y^{2} + 30 x - 13 y - 25 = 0 \end{array}$

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Definition of a circle

Remember concepts with our Masterclasses.

The equation of a circle passing through (1, 1) and points of intersection of x2+y2+13x−3y=0 and 2x2+2y2+4x−7y−25=0 is

The equation of the required circle isx2+y2+13x−3y+λx2+y2+2x−72y−252=0This passes through (1, 1). ∴ 12+λ(−12)=0⇒λ=1.Hence, the required circle is x2+y2+13x−3y+x2+y2+2x−72y−252=0⇒4x2+4y2+30x−13y−25=0

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The equation of a circle passing through $(1, 1)$ and points of intersection of $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is