Question

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

Answer 1

inside the circle

Explanation:

The equation of a circle centred at the origin is

x² + y² = r² ( r is the radius )

x² + y² = 25 ← is in this form

with r =
`\sqrt{25` = 5

Calculate the distance d from the centre to the point (- 4, 2 ) using the distance formula

d =
`\sqrt{(x_(2)-x_(1))^2+(y_(2)-y_(1))^2 }`

with (x₁, y₁ ) = (0, 0) and (x₂, y₂ ) = (- 4, 2)

d=
`√((-4-0)^2+(2-0)^2)`

=
`√((-4)^2+2^2)`

=
`√(16+4)`

=
`√(20)`

≈ 4.5 ( to 1 dec. place )

Since 4.5 is less than the radius of 5

Then (- 4, 2 ) lies inside the circle

Answer 2

Given equation of the Circle is ,

`\sf\implies x^2 + y^2 = 25`

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 ,

`\sf\implies (x-0)^2 +( y-0)^2 = 5 ^2`

Here we can say that ,

• Radius = 5 units

• Centre = (0,0)

Finding distance between the two points :-

`\sf\implies Distance = √( (0+4)^2+(2-0)^2) \\\\\sf\implies Distance = √( 16 + 4 ) \\\\\sf\implies Distance =√(20)\\\\\sf\implies\red{ Distance = 4.47 }`

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

Hence the point lies within the circle .