Prove that the straight lines `x/a-y/b =m and x/a+y/b=1/m,` where `a and b` are `'m'` is a parameter, always meet on agiven positive real numbers and hyperbola.

Prove that the locus of point of intersection of the straight lines (tx)/(a)-(y)/(b)+t=0 and (x)/(a)+(ty)/(b)-1=0 where t is a parameter is an ellipse.Prove also they meet at the point whose eccentric angle is 2tan^(-1)t

The differential equation of the family of straight line y=mx+(4)/(m) , where m is the parameter, is

x^(m)-:y^(m)=(x-:y)^(m) , where x and y are non zero rational numbers and m is a positive integer.

Prove that the tangents to family of hyperbola (x^(2))/(b^(2)+lambda)-(y^(2))/(b^(2))=1 (where lambda is a real parameter) at their point of intersection with x^(2)-y^(2)=2b^(2)+lambda^(2) are a pair of fixed lines. Find the equations of the lines.

Prove that the straight line y=mx+2c sqrt(-m),m in R, always touches the hyperbola xy=c^(2)

Prove that the straight lines x/alpha=y/beta=z/gamma,x/l=y/m=z/n and x/(a alpha)=y/(b beta)=z/(c gamma) will be co planar if l/alpha(b-c)+m/beta(c-a)+n/gamma(a-b)=0

A variable straight line is drawn through the point of intersection of the straight lines (x)/(a)+(y)/(b)=1 and (x)/(b)+(y)/(a)=1 and meets the coordinate axes at A and B. Show that the locus of the midpoint of AB is the curve 2xy(a+b)=ab(x+y)