Q If two curves axâ€‹2+2hxy+by2+2gx+2fy+c = 0 and a'x2+2h'xy+b'y2+2g'x+2f ' y +c ' = 0 intersect in four concyclic - Maths - Conic Sections

Question

Q If two curves
axâ€‹2+2hxy+by2+2gx+2fy+c =
0 and a'x2+2h'xy+b'y2+2g'x+2f ' y +c ' = 0
intersect in four concyclic points then prove that a-b/h = a'-b'/h'

Aarushi Mishra · Accepted Answer

C1: ax2+by2+2hxy+2gx+2fy+c=0C2: a'x2+b'y2+2h'xy+2g'x+2f'y+c'=0Using family of curves, A curve passing through intersection of C1 and C2 will be given byax2+by2+2hxy+2gx+2fy+c + λ(a'x2+b'y2+2h'xy+2g'x+2f'y+c')=0λa'+ax2+λb'+by2+2λh'+hxy+2λg'+gx+2λf'+λy+λc'+c=0Since C1 and C2 intersect at four concyclic points, therefore it is possible to get equation of a cricle passing through these points using family of curvesIn general equation for a cricle coefficient of xy is zeroλh'+h=0λ=-hh'Also in general equation of a circle cofficient of x2 and y2 are equalλa'+a=λb'+bputting value of λ-hh'a'+a=-hh'b'+bhh'a'-hh'b'=a-bhh'a'-b'=a-ba'-b'h'=a-bh

Q. If two curves ax​2+2hxy+by2+2gx+2fy+c = 0 and a'x2+2h'xy+b'y2+2g'x+2f ' y +c ' = 0 intersect in four concyclic points then prove that a-b/h = a'-b'/h'

Q. If two curves ax²+2hxy+by²+2gx+2fy+c = 0 and a'x²+2h'xy+b'y²+2g'x+2f ' y +c ' = 0 intersect in four concyclic points then prove that a-b/h = a'-b'/h'