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

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


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

  1. 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]\begin{lgathered}\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 }\end{lgathered} [/tex]

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

    Hence the point lies within the circle.

    Reply
  2. Answer:

    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]

    step2: 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

    Reply

Leave a Comment