Answered

Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.

does the point (-4, 2) lie inside or outside or on the circle x^2 + y^2 = 25?​
​

Sagot :

VirαtKσhli

Given equation of the Circle is ,

[tex]\sf\implies x^2 + y^2 = 25 [/tex]

And we need to tell that whether the point (-4,2) lies inside or outside the circle. On converting the equation into Standard form and determinimg the centre of the circle as ,

[tex]\sf\implies (x-0)^2 +( y-0)^2 = 5 ^2[/tex]

Here we can say that ,

• Radius = 5 units

• Centre = (0,0)

Finding distance between the two points :-

[tex]\sf\implies Distance = \sqrt{ (0+4)^2+(2-0)^2} \\\\\sf\implies Distance = \sqrt{ 16 + 4 } \\\\\sf\implies Distance =\sqrt{20}\\\\\sf\implies\red{ Distance = 4.47 }[/tex]

Here we can see that the distance of point from centre is less than the radius.

Hence the point lies within the circle .

Nayefx

inside the circle

Step-by-step explanation:

we want to verify whether (-4,2) lies inside or outside or on the circle to do so recall that,

  1. if [tex]\displaystyle (x-h)^2+(y-k)^2>r^2[/tex] then the given point lies outside the circle
  2. if [tex]\displaystyle (x-h)^2+(y-k)^2<r^2[/tex] then the given point lies inside the circle
  3. if [tex]\displaystyle (x-h)^2+(y-k)^2=r^2[/tex] then the given point lies on the circle

step-1: define h,k and r

the equation of circle given by

[tex] \displaystyle {(x - h)}^{2} + (y - k) ^2= {r}^{2} [/tex]

therefore from the question we obtain:

  • [tex] \displaystyle h= 0[/tex]
  • [tex] \displaystyle k= 0[/tex]
  • [tex] {r}^{2} = 25[/tex]

step-2: verify

In this case we can consider the second formula

the given points (-4,2) means that x is -4 and y is 2 and we have already figured out h,k and r² therefore just substitute the value of x,y,h,k and r² to the second formula

[tex] \displaystyle {( - 4 - 0)}^{2} + (2 - 0 {)}^{2} \stackrel {?}{ < } 25[/tex]

simplify parentheses:

[tex] \displaystyle {( - 4 )}^{2} + (2 {)}^{2} \stackrel {?}{ < } 25[/tex]

simplify square:

[tex] \displaystyle 16 + 4\stackrel {?}{ < } 25[/tex]

simplify addition:

[tex] \displaystyle 20\stackrel { \checkmark}{ < } 25[/tex]

hence,

the point (-4, 2) lies inside the circle

Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.