Question

find the value of \( a \) if \( a x^{2}+9 y^{2}-3 x+2 y-1=0 \) Reprents on a circle and find its radius.

Answer 1

Given equation is :

ax² + 9y² - 3x + 2y -1 = 0 .....(1)

Equation (1) is represent an equation of circle if coefficient of x² = coefficient of y².

∴ a = 9 then equation (1) represents an equation of circle.

Then equation (1) becomes

9x² + 9y² - 3x + 2y -1 = 0

⇒ x² + y² - \(\frac{3}{9}x\) + \(\frac{2}{9}y\) - \(\frac{1}{9}\) = 0

⇒ x² - \(\frac{3}{9}x\) + (\(\frac{3}{18}\))² + y² + \(\frac{2}{9}y\) + \((\frac{1}{9})^2\)-  \(\frac{1}{9}\) -  (\(\frac{3}{18}\))² - \((\frac{1}{9})^2\) = 0

⇒ (x - \(\frac{3}{18}\))² + (y + \(\frac{1}{9}\))² = \(\frac{36+9+4}{18^2}\) 

⇒  (x - \(\frac{3}{18}\))² + (y + \(\frac{1}{9}\))² = \(\frac{49}{18^2}\) 

= \((\frac{7}{18})^2\)

∴ Radius of the formed circle is \(\frac{7}{18}\).