Given point A (a, b), B (-a, -b) and curve C lie in the xOy plane. Point P moves along the curve C. If the product of the gradient of the PA line and the PB gradient line is always equal to the k constant, then C is a circle if k?

Question

332 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-01-17T09:56:23

#k=-1#

Let the point be #P(x,y)#.

As #A(a,b)#, slope of #PA# is #(y-b)/(x-a)# and as #B(-a,-b)#, slope of #PB# is #(y+b)/(x+a)#

Now product of the two slopes is #k#, hence

#(y-b)/(x-a)xx(y+b)/(x+a)=k#

or #y^2-b^2=kx^2-ka^2#

or #kx^2-y^2=ka^2-b^2#

Now in a circle coefficients of #x^2# and #y^2# areequal, hence #k=-1# and equation of circle is

#x^2+y^2=b^2-a^2#