A variable line moves in such a way that the length of perpendiculars p1 and p2 from two fixed points (a,0) and (-a,0)
is connected as p12 - p22=k(constant)then the line touches a fixed
A)circle B)parabola
C)hyperbola D)ellipse
Let the line be y=mx+c or mx-y+c=0.
We have, p12 - p22=k.
\(\Rightarrow \left({\frac{c+am}{\sqrt{1+m^2}}\right)^2-\left({\frac{c-am}{\sqrt{1+m^2}}\right)^2=k\)
\(\Rightarrow 2c.2am=(1+m^2)k\)
\(\Rightarrow c=\frac{k}{4a}(m+\frac{1}{m})\)
So, the equation of the line is:
\(y=mx+\frac{k}{4a}(m+\frac{1}{m})\)
Or, \(y=m(x+\frac{k}{4a})+\frac{k}{4am}\)
Or, \(y=m(x+A)+\frac{A}{m}\)
Or, \(y=mX+\frac{A}{m}\) (shift of origin)
thank you sir!
qn2).a rectangular hyperbola passes through the point (0,0) and has one of its focus at (1,0), thge latus rectum is 2 units. the locus of the other focus is:-