What is the position of the point P (2, 4) with respect to the circle S: x2 + y2 - 6x - 10y + 26 = 0 ?

CONCEPT:

If S(x, y) : x2 + y2 + 2gx + 2fy + c = 0 is a circle and P (x1 , y1) is a point then the position of the point P with respect to the circle can be:

• P lies on the circle S if S(x1, y1) = 0
• P lies inside the circle S if S(x1, y1) < 0
• P lies outside the circle if S(x1, y1) > 0

CALCULATION:

Given: Equation of circle is S: x2 + y2 - 6x - 10y + 26 = 0 and the point P (2, 4)

Here, x1 = 2, y1 = 4

First let's find out S(x1 , y1)

⇒ S(x1 , y1) = x12 + y12 - 6x1 - 10y1 + 26 = 0

⇒ S(2, 4) = 4 + 16 - 12 - 40 + 26 = - 6 < 0

As we know that, if S(x1, y1) < 0 for point P (x1 , y1) then the point P lies inside the circle S.

Hence, option A is the correct answer.