Question

A circle whose center is at (-6, 8) passes through the origin, which of the following points are not on the circle?
$
  {\text{A}}{\text{. }}\left( { - 2,12} \right) \\
  {\text{B}}{\text{. }}\left( { - 16,8} \right) \\
  {\text{C}}{\text{. }}\left( { - 6, - 2} \right) \\
  {\text{D}}{\text{. }}\left( {4,8} \right) \\
$

Answer 1

Hint: To solve the question we find the radius of the circle using the formula for distance between two points. (One point is the origin and the other is the center.)
Then we find the distance from each of the given points from the center.

Complete step-by-step answer:
Given Data:
The circle passes through the origin.
The line joining origin and center of the circle is the radius of the circle.
We know in a ∆ABC, Pythagoras theorem states that the hypotenuse ${\text{A}}{{\text{C}}^2} = {\text{A}}{{\text{B}}^2} + {\text{B}}{{\text{C}}^2}$
The distance between two points with coordinates $\left( {{{\text{x}}_1},{{\text{y}}_1}} \right){\text{ and }}\left( {{{\text{x}}_2},{{\text{y}}_2}} \right){\text{ is d = }}\sqrt {{{\left( {{{\text{x}}_2} - {{\text{x}}_1}} \right)}^2} + {{\left( {{{\text{y}}_2} - {{\text{y}}_1}} \right)}^2}} $

Observing the figure, we get a right triangle with points at origin O, (0, 8) and (-6, 8) and with radius as the hypotenuse of this triangle.
∴ ${{\text{r}}^2}$ = ${\left( { - 6} \right)^2} + {8^2}$​ (Using Pythagoras Theorem)
          = 36 + 64
⟹${{\text{r}}^2}$ = 100​
⟹r = 10


To check whether the given points lie on the circle, we will check their distance from the center.
Let the distance from the point be ‘d’.

A.Distance of point (−2, 12) from the center (−6, 8) by applying distance formula.
Distance (d) =$\sqrt {{{\left( {8 - 12} \right)}^2} + {{\left( { - 6 + 2} \right)}^2}} = \sqrt {32} = 5.66{\text{ units}}$ ​
d < r , which does not form the radius.
Hence, (−2, 12) does not lie on the circle. It lies inside the circle.

B.Distance of point (−16, 8) from the center (−6, 8) by applying distance formula.
Distance (d) = $\sqrt {{{\left( {8 - 8} \right)}^2} + {{\left( { - 6 + 16} \right)}^2}} = \sqrt {100} = 10{\text{ units}}$
d = r, which forms the radius.
Hence, (−16, 8) lies on the circle.

C.Distance of point (−6, −2) from the center (−6, 8) by applying distance formula.
Distance (d) = $\sqrt {{{\left( {8 + 2} \right)}^2} + {{\left( { - 6 + 6} \right)}^2}} = \sqrt {100} = 10{\text{ units}}$
d = r, which forms the radius.
Hence, (−6, −2) lies on the circle.

D.Distance of point (4, 8) from the center (−6, 8) by applying distance formula.
Distance (d) = $\sqrt {{{\left( { - 6 - 4} \right)}^2} + {{\left( {8 - 8} \right)}^2}} = \sqrt {100} = 10{\text{ units}}$
d = r, which forms the radius.
Hence, (4, 8) lies on the circle.

So, point A does not lie on the circle.
Hence Option A is the correct answer.

Note: The key in such problems is to know how to determine if a point is said to be on the circle.
If the distance from the point d,
d > r (point is outside the circle)
d =r (point is on the circle)
d < r (point is inside the circle).