Question

A circle is centered at H (4,0) and has a radius of 10. Where does the point L(-2,8) lie? (khan)

a. inside the circle
b. on the circle
c. outside the circle

Answer 1

b) on the circle

The point lies on the circle S =0

Explanation:

Step(i):-

Given center of the circle = H(4,0)

Radius of the circle 'r' = 10

Equation of the circle

`(x -h)^(2) +(y-k)^(2) = r^(2)`

`(x -4)^(2) +(y-0)^(2) = 10^(2)`

`(x )^(2) -8 x + 16 + (y)^(2) = 10^(2)`

`(x )^(2) -8 x + (y)^(2) = 10^(2) - 16`

`x ^(2) -8 x + y^(2) = 84`

Step(ii):-

S =0 is a circle and P(x₁ , y₁) be a point in its plane

Then (i) P lies inside the circle S = 0 ⇔ S₁₁ < 0

ii) P lies outside the circle S = 0 ⇔ S₁₁ > 0

iii) P lies on the circle S = 0 ⇔ S₁₁ = 0

now

`S_(11) = x^(2) _(1) + y^(2) _(1) + 2 g x_(1) + 2 f y_(1) +c`

`S_(11) = x_(1) ^(2) -8 x_(1) + y_(1) ^(2) = 84`

Given point ( -2 , 8)

`S_(11) = (-2) ^(2) -8 (-2) + (8) ^(2) -84 = 0`

P lies on the circle S = 0 ⇔ S₁₁ = 0