If equation \(a x^2+2 h x y+b y^2+2 g x+2 f y+c=0\) represents a rectangular hyperbola, then

If equation \(a x^2+2 h x y+b y^2+2 g x+2 f y+c=0\) represents a rectangular hyperbola, then
  1. \(\Delta \neq 0, \mathrm{~h}^2<\mathrm{ab}, \mathrm{a}+\mathrm{b} \neq 0\)
  2. \(\Delta \neq 0, \mathrm{h}^2>\mathrm{ab}, \mathrm{a}+\mathrm{b}=0\)
  3. \(\Delta \neq 0, \mathrm{h}^2=\mathrm{ab}, \mathrm{a}+\mathrm{b}=0\)
  4. \(\Delta \neq 0, \mathrm{~h}^2<\mathrm{ab}, \mathrm{a}+\mathrm{b}=0\)

Solution

It is fundamental concept that for equation
\(a x^2+2 h x y+b y^2+2 g x+2 f y+c=0\) to represents a rectangular hyperbola it has to satisfy following condition,
\(\Delta=\mathrm{abc}+2 \mathrm{fgh}-\mathrm{af}^2-\mathrm{bg}^2-\mathrm{ch}^2 \neq 0, \mathrm{h}^2>\mathrm{ab}\) and \(\mathrm{a}+\mathrm{b}=0\)

Asked in: MHT CET 2020 (15 Oct Shift 1)

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