How to find the relative position of a point with respect to a conic ax2 +by2 + 2hxy +2gx +2fy + c = 0 ?

Dear Pratik

f(x,y)=ax2 +by2 + 2hxy +2gx +2fy + c = 0

first you should find out conic section by compairing it with standard equation

3 position is possible for a point w.r.t conic .one is inside the conic ,second is outside the conic and third is on the conic.

if f(x1,y1) =0 then point lie on the conic.

if f(x1,x2) and f(x2,y2)   are of same sign the both point lie either inside or outside of the conic,

if f(x1,x2) and f(x2,y2) are of apposite sign then one point lie inside the conic and other point lie outside the conic,

In some books i saw that such equtions are first converted into the general eq. of that specific conic i.e, if it is parabola than it is convertd in the form of y2 = 4ax and after the points are substituted. And such substitution changes the whole answer

Dear Pratik

Answer does not change either you solve it in general form or solve it after converting in to standard form.

if you want you can convert first in to standard form and then find out the relative position.

suppose standard form comes as y²=4ax

say S =y²-4ax

now put point (x1,y1)

S1 =y1² -4ax1

if S1 < 0 then point lie inside the parabola

if S1>0 then point lie out side the parabola.

if you have still doubt in both method then put the example i will compair both the method in that example.

but put that question in new post.