Question

The equation $\text{k}\left({\text{x}}^2+{\text{y}}^2\right)-\text{x - y + k = 0}$ represents a real circle, if
$$\begin{gathered} \text{A}\text{. k < }\sqrt{2} \\ \text{B}\text{. k > }\sqrt{2} \\ \text{C}\text{.|k| < }{\displaystyle \frac{1}{\sqrt{2}}} \\ \text{D}\text{. 0 < |K| }⩽{\displaystyle \frac{1}{\sqrt{2}}} \end{gathered}$$

Answer 1

Hint: To check if the equation represents a circle, we transform the given circle equation into the general form of a circle. We then use the condition that represents a real circle to verify.

Complete step-by-step answer:
We know, for an equation ${\text{x}}^2+{\text{y}}^2+2\text{gx + 2fy + c = 0}$ to represent a real circle, the condition is $\sqrt{\left({\text{g}}^2+{\text{f}}^2-\text{c}\right)}>0$.

Given equation $\text{k}\left({\text{x}}^2+{\text{y}}^2\right)-\text{x - y + k = 0}$
We divide the equation by k and compare it with the circle equation we get,
${\text{x}}^2+{\text{y}}^2-{\displaystyle \frac{1}{\text{k}}}\text{x - }{\displaystyle \frac{1}{\text{k}}}\text{y + 1 = 0}$
$\Rightarrow \text{g = - }{\displaystyle \frac{1}{2\text{k}}}\text{ , f = - }{\displaystyle \frac{1}{2\text{k}}}\text{ , c = 1}$

To represent a circle, $\sqrt{\left({\text{g}}^2+{\text{f}}^2-\text{c}\right)}>0$ must hold true,

$$\begin{gathered} \Rightarrow \sqrt{\left({\left(-{\displaystyle \frac{1}{2\text{k}}}\right)}^2+{\left(-{\displaystyle \frac{1}{2\text{k}}}\right)}^2-1\right)}>0 \\ \Rightarrow \sqrt{{\displaystyle \frac{1}{4{\text{k}}^2}}+{\displaystyle \frac{1}{4{\text{k}}^2}}-1}\text{ > 0} \\ \Rightarrow \sqrt{{\displaystyle \frac{1}{2{\text{k}}^2}}-1}\text{ > 0} \\ \Rightarrow {\displaystyle \frac{1}{2{\text{k}}^2}}\text{ > 1} \\ \Rightarrow \text{1 > 2}{\text{k}}^2 \\ \Rightarrow {\displaystyle \frac{1}{2}}>{\text{k}}^2 \\ \Rightarrow |\text{k| < }{\displaystyle \frac{1}{\sqrt{2}}} \end{gathered}$$

Hence, for the equation $\text{k}\left({\text{x}}^2+{\text{y}}^2\right)-\text{x - y + k = 0}$ represents a real circle if $\text{|k| < }{\displaystyle \frac{1}{\sqrt{2}}}$.
Option C is the right answer.

Note: In order to solve such types of questions the key is to have adequate knowledge of the general equation and conditions for a circle to represent a real circle. And then we modify the given circle into the form of a general equation of a circle and use the required condition, we have to be very careful while comparing the given equation with the general form of circle to obtain the values of g and f.