Question

x2+y2+8x+12y+15=0 if the abscissa of points A,B are the roots of the equation x2+2ax-b2=0 and ordinates of A,B are roots of y2+2py-q2=0 then find the equation of the circle for which AB is a diameter

Answer 1

Answer:

Let coordinates of A and B are (x1,y1) and (x2,y2).

x1 and x2 are the roots of x^2 +2ax -b^2=0, therefore

x1+x2= -2a……………(1). and. x1.x2= -b^2……………..(2)

y1 ,y2 are the roots of y^2+2py-q^2=0 , therefore

y1+y2=-2p……………..(3). and. y1.y2=-q^2……………..(4)

Length of diameter AB =√[(x1-x2)^2+(y1-y2)^2]

Formula used. [(a-b)^2=(a+b)^2–4a.b]

Diameter AB =√[(x1+x2)^2–4x1.x2+(y1+y2)^2–4y1.y2]

= √[4a^2+4b^2+4p^2+4q^2]

= 2.√(a^2+b^2+p^2+q^2) units.

Radius = diameter/2 = 2.√(a^2+b^2+p^2+q^2)/2

Radius =√(a^2+b^2+p^2+q^2). units. Answer