If the point $\left( \lambda ,-\lambda \right) $ lies inside the circle $x^{2}+y^{2}-4x+2y-8=0$, then find the range of $\lambda$.

Hint: In this question it is given that if a point $\left( \lambda ,-\lambda \right) $ lies inside the circle $x^{2}+y^{2}-4x+2y-8=0$, then we have to find the range of $\lambda$. So to find the solution we have to know that if any point (a,b) lies inside the circle then we can write, $a^{2}+b^{2}-4a+2b-8<0$ …………….(1).

Complete step-by-step answer:
Since it is given that point $\left( \lambda ,-\lambda \right) $ lies inside the circle.
So by (1) we can write,
$$a^{2}+b^{2}-4a+2b-8<0$$
$$\Rightarrow \lambda^{2} +\lambda^{2} -4\lambda +2\left( -\lambda \right) -8<0$$
$$\Rightarrow 2\lambda^{2} -4\lambda -2\lambda -8<0$$
$$\Rightarrow 2\lambda^{2} -6\lambda -8<0$$
$$\Rightarrow 2\left( \lambda^{2} -3\lambda -4\right) <0$$
$$\Rightarrow \left( \lambda^{2} -3\lambda -4\right) <0$$ [dividing both side by 2]
Now by middle term factorisation,
$$\Rightarrow \lambda^{2} -4\lambda +\lambda -8<0$$
$$\Rightarrow \lambda \left( \lambda -4\right) +1\left( \lambda -4\right) <0$$
Now by taking $\left( \lambda -4\right) $ common,
$$\Rightarrow \left( \lambda -4\right) \left( \lambda +1\right) <0$$........(2)
Whenever multiplication of two terms less than zero, i.e ab<0, then
Either a<0 and b>0,
Or, a>0 and b<0,

So by this we can write (2) as,
Either, $\left( \lambda -4\right) <0$ and $\left( \lambda +1\right) >0$
Which implies, $\lambda <4$ and $\lambda >-1$.
So the range of $\lambda$ is $-1<\lambda <4$ .
Also this can be written as $\lambda \in \left( -1,4\right) $


Or, $\left( \lambda -4\right) >0$ and $\left( \lambda +1\right) <0$
Which implies, $\lambda >4$ and $\lambda <-1$
So there is no common region by this above condition.
Thus the required solution is $\lambda \in \left( -1,4\right) $.


Note: To solve this type of question you should keep in mind that, choose proper conditions according to the position of a point. And also while solving any inequation always remember that whenever multiplication of two terms less than zero, i.e ab<0, then either, a<0 and b>0, Or, a>0 and b<0. And for sets or intervals the meaning of “and” is intersection.