find the value of \( a \) if \( a x^{2}+9 y^{2}-3 x+2 y-1=0 \) Reprent

1.	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» 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 \(\frac{7}{18}\).

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»

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 \(\frac{7}{18}\).

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