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

Does the point (-2.5, 3.5) lie inside, outside or on the circle x2+y2=20{{x}^{2}}+{{y}^{2}}=20?

Explanation

Solution

Hint: Put this point (-2.5, 3.5) in x2+y2=20{{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 x2+y2−r2{{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 x2+y2−r2{{x}^{2}}+{{y}^{2}}-{{r}^{2}} and then see what kind of values are obtaining.
If after substituting the point (a, b) in x2+y2−r2{{x}^{2}}+{{y}^{2}}-{{r}^{2}} the answer is 0 then the point (a, b) lies on the circle.
S=x2+y2−r2 ⇒S=a2+b2−r2=0 \begin{aligned} & S={{x}^{2}}+{{y}^{2}}-{{r}^{2}} \\\ & \Rightarrow S={{a}^{2}}+{{b}^{2}}-{{r}^{2}}=0 \\\ \end{aligned}
If after substituting the point (a, b) in x2+y2−r2{{x}^{2}}+{{y}^{2}}-{{r}^{2}} the answer is less than 0 then the point (a, b) lies inside the circle.
S=x2+y2−r2 ⇒S=a2+b2−r2<0 \begin{aligned} & S={{x}^{2}}+{{y}^{2}}-{{r}^{2}} \\\ & \Rightarrow S={{a}^{2}}+{{b}^{2}}-{{r}^{2}}<0 \\\ \end{aligned}
If after substituting the point (a, b) in x2+y2−r2{{x}^{2}}+{{y}^{2}}-{{r}^{2}} the answer is greater than 0 then the point (a, b) lies outside the circle.
S=x2+y2−r2 ⇒S=a2+b2−r2>0 \begin{aligned} & S={{x}^{2}}+{{y}^{2}}-{{r}^{2}} \\\ & \Rightarrow S={{a}^{2}}+{{b}^{2}}-{{r}^{2}}>0 \\\ \end{aligned}
Now, using the above conditions we are going to find the position of point (-2.5, 3.5) with respect to the circle x2+y2−20{{x}^{2}}+{{y}^{2}}-20
Let us assume that x2+y2−20{{x}^{2}}+{{y}^{2}}-20 is equal to S1.
Now, substitute the point (-2.5, 3.5) inx2+y2−20{{x}^{2}}+{{y}^{2}}-20we get,
(−2.5)2+(3.5)2−20 =6.25+12.25−20 =18.50−20 =−1.50 \begin{aligned} & {{\left( -2.5 \right)}^{2}}+{{\left( 3.5 \right)}^{2}}-20 \\\ & =6.25+12.25-20 \\\ & =18.50-20 \\\ & =-1.50 \\\ \end{aligned}
As we can see from the above that after substituting the point (-2.5, 3.5) in x2+y2−20{{x}^{2}}+{{y}^{2}}-20 the answer is less than 0 or (S1<0)\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 x2+y2=20{{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.