Question

Prove that x² + y² + 2gx + 2fy + C = 0 represents a circle. Also, find its centre and radius.

➡️ don't spam ✖️​

Answer 1

$\large\bf{\underline{\underline{Question:-}}}$

Prove that x² + y² + 2gx + 2fy + C = 0 represents a circle. Also, find its centre and radius.

$\large\bf{\underline{\underline{Solution:-}}}$

We have,

x² + y² + 2gx + 2fy + C = 0

$\implies$ x² + y² + 2gx + 2fy = - C

$\implies$ (x² + 2gx) + (y² + 2fy) = -C

$\implies$ (x² + 2.x.g + g²) + (y² + 2.y.f + f²) = g² + f² - C

$\implies$ (x + g)² + (y + f)² = g² + f² - C

$\implies$ {x - (-g)}² + {y - (-f)}² = $\sqrt{g² + f² - C}^{2}$

which is the form of (x - h)² + (y - k)² = r²

This shows that x² + y² + 2gx + 2fy = 0 represents a circle.

This type of equation of a circle is called general equation of the circle.

Hence, the centre and radius of the circle are (-g, -f) and $\sqrt{g² + f² -C}$

$\pink{Hope \: it \: helps}$

Answer 2

$\large\bf{\underline{\underline{Question:−}}}$

Prove that x² + y² + 2gx + 2fy + C = 0 represents a circle. Also, find its centre and radius.

$\large\bf{\underline{\underline{Solution:-}}}$

We have,

x² + y² + 2gx + 2fy + C = 0

⟹ x² + y² + 2gx + 2fy = - C

⟹ (x² + 2gx) + (y² + 2fy) = -C

⟹ (x² + 2.x.g + g²) + (y² + 2.y.f + f²) = g² + f² - C

⟹ (x + g)² + (y + f)² = g² + f² - C

$⟹ {x - (-g)}² + {y - (-f)}² = \sqrt{g² + f² - C}^{2} </p><p>g²+f²−C$

which is the form of (x - h)² + (y - k)² = r²

This shows that x² + y² + 2gx + 2fy = 0 represents a circle.

This type of equation of a circle is called general equation of the circle.

$Hence, the \: centre \: and \: radius \: of \: the \: circle \: are (-g, -f) and \sqrt{g² + f² -C}$