Question

The equation of the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyperbola xy = c2 is :-

Answer 1

Well, you can find out equation by using basic rules of Geometry,
Because two points (x₁, y₁) and (x₂, y₂) are given then,
Slope of chord is (y₂ - y₁)/(x₂ - x₁)
Now, equation of chord is (y - y₁) = (y₂ - y₁)/(x₂ - x₁)(x - x₁)

But we have to use term c in equation of chord.
so, let's start .
can we write (x₁, y₁) = (x₁, c²/x₁) [ ∵ x₁y₁ = c² ⇒ y₁ = c²/x₁ ]
similarly , (x₂,y₂) = (x₂, c²/x₂)

Now, slope of chord is c²(x₁ - x₂)/x₁x₂(x₂ -x₁) = -c²/x₁x₂
So, equation of chord is given by
(y - c²/x₁) + c²/x₁x₂(x - x₁) = 0
y - c²/x₁ + c²/x₁x₂.x - c²/x₂ = 0
x₁x₂y + c²x = (x₁ + x₂)c²