Question

The equation of the circle passing through the point (1, 1) and having two diameters along the pair of lines x² − y² −2x + 4y − 3 = 0, is

Options

• x² + y² − 2x − 4y + 4 = 0

•  x² + y² + 2x + 4y − 4 = 0

• x² + y² − 2x + 4y + 4 = 0

• none of these

Answer 1

x² + y² − 2x − 4y + 4 = 0
Let the required equation of the circle be \[\left( x - h \right)^2 + \left( y - k \right)^2 = a^2\] .

Comparing the given equation x² − y² −2x + 4y − 3 = 0 with \[a x^2 + b y^2 + 2hxy + 2gx + 2fy + c = 0\] ,we get:

\[a = 1, b = - 1, h = 0, g = - 1, f = 2, c = - 3\]

Intersection point = \[\left( \frac{hf - bg}{ab - h^2}, \frac{gh - af}{ab - h^2} \right)\] = \[\left( \frac{- 1}{- 1}, \frac{- 2}{- 1} \right) = \left( 1, 2 \right)\]

Thus, the centre of the circle is \[\left( 1, 2 \right)\] .

The equation of the required circle is \[\left( x - 1 \right)^2 + \left( y - 2 \right)^2 = a^2\] .

Since circle passes through (1, 1), we have:

\[1 = a^2\]

∴ Equation of the required circle:

\[\left( x - 1 \right)^2 + \left( y - 2 \right)^2 = 1\]

\[\Rightarrow\] \[x^2 + y^2 - 2x - 4y + 4 = 0\]