Mathematics

Problem

For the circle

x^2+y^2+2gx+2fy+c=0

an algebraic expression

S_1

is often evaluated for a test point

P(x_1,y_1)

. Answer the following questions. a. (i) State the explicit expression for

S_1

in terms of

x_1,y_1,g,f

and

c

. b. (ii) Explain, without using the distance formula, why the sign of

S_1

determines whether

P

lies inside, on, or outside the circle. c. (iii) For the circle

x^2+y^2-4x+6y-3=0

and the point

P(1,2)

, evaluate

S_1

and, on the basis of its sign, classify the position of

P

with respect to the circle.

Intuitive explanation

The key insight is that $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$ tells you the "signed distance" from point $P$ to the circle—it's what you get when you substitute the point's coordinates into the circle equation. If $S_1 = 0$ , the point lies on the circle; if $S_1 > 0$ , it's outside; if $S_1 < 0$ , it's inside. This single expression unlocks whether a point is inside, on, or outside without needing to compute the actual radius or center explicitly.

Intuitive explanation

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