Answered

Get the answers you need from a community of experts on IDNLearn.com. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.

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

Sagot :

VirαtKσhliidn

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 .

Nayefxidn

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

We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thanks for visiting IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more helpful information.