Question

Does the point (-2.5, 3.5) lie inside, outside or on the circle ${{x}^{2}}+{{y}^{2}}=20$?

Answer 1

Hint: Put this point (-2.5, 3.5) in ${{x}^{2}}+{{y}^{2}}=20$. And if we get an answer equal to 0 then point lies on the circle, if the answer is less than 0 then point lies inside the circle and if the answer is greater than 0 then point lies outside the circle.

Complete step-by-step answer:
Let us assume that ${{x}^{2}}+{{y}^{2}}-{{r}^{2}}$as S.
We can find the position of a point (a, b) whether it lies inside, outside or on the circle by substituting the point in ${{x}^{2}}+{{y}^{2}}-{{r}^{2}}$ and then see what kind of values are obtaining.
If after substituting the point (a, b) in ${{x}^{2}}+{{y}^{2}}-{{r}^{2}}$ the answer is 0 then the point (a, b) lies on the circle.
$\begin{align}
  & S={{x}^{2}}+{{y}^{2}}-{{r}^{2}} \\
 & \Rightarrow S={{a}^{2}}+{{b}^{2}}-{{r}^{2}}=0 \\
\end{align}$
If after substituting the point (a, b) in ${{x}^{2}}+{{y}^{2}}-{{r}^{2}}$ the answer is less than 0 then the point (a, b) lies inside the circle.
$\begin{align}
  & S={{x}^{2}}+{{y}^{2}}-{{r}^{2}} \\
 & \Rightarrow S={{a}^{2}}+{{b}^{2}}-{{r}^{2}}<0 \\
\end{align}$
If after substituting the point (a, b) in ${{x}^{2}}+{{y}^{2}}-{{r}^{2}}$ the answer is greater than 0 then the point (a, b) lies outside the circle.
$\begin{align}
  & S={{x}^{2}}+{{y}^{2}}-{{r}^{2}} \\
 & \Rightarrow S={{a}^{2}}+{{b}^{2}}-{{r}^{2}}>0 \\
\end{align}$
Now, using the above conditions we are going to find the position of point (-2.5, 3.5) with respect to the circle ${{x}^{2}}+{{y}^{2}}-20$
Let us assume that ${{x}^{2}}+{{y}^{2}}-20$ is equal to S1.
Now, substitute the point (-2.5, 3.5) in${{x}^{2}}+{{y}^{2}}-20$we get,
$\begin{align}
  & {{\left( -2.5 \right)}^{2}}+{{\left( 3.5 \right)}^{2}}-20 \\
 & =6.25+12.25-20 \\
 & =18.50-20 \\
 & =-1.50 \\
\end{align}$
As we can see from the above that after substituting the point (-2.5, 3.5) in ${{x}^{2}}+{{y}^{2}}-20$ the answer is less than 0 or $\left( {{S}_{1}}<0 \right)$ so the point lies inside the circle.
Hence, from the above solution we say that the point (-2.5, 3.5) lies inside the circle ${{x}^{2}}+{{y}^{2}}=20$.

Note: The condition to find the position of a point with respect to ellipse and parabola is the same as we have shown above for a circle but for hyperbola there is a slight change in the condition.
For hyperbola, the condition for a point to lie inside the hyperbola is S > 0 and a point to lie outside the hyperbola is S < 0 while the condition for a point to lie on the hyperbola is the same as that of a circle.