For the circle an algebraic expression is often evaluated for a test point

MATHEMATICS · Multi-part · CBSE · Mathematics · Conic Sections · Conic Sections

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. (i) State the explicit expression for

S_1

in terms of

x_1,y_1,g,f

and

c

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

S_1

determines whether

P

lies inside, on, or outside the circle. (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.