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

Question

The equation of the chord joining two points (x1, y1) and (x2, y2) on the rectangular hyperbola xy = c2 is (A) (x/x1+x2)+(y/y1+y2)=1 (B) (x/x1-x2)+(y/y1-y2)=1 (C) (x/y1+y2)+(y/x1+x2)=1 (D) (x/y1-y2)+(y/x1-x2)=1

Tardigrade Admin · Accepted Answer

The mid -point of the chord is ((x1+x2/2), (y1+y2/2)) The equation of the chord in terms if its mid-point is T=S1, i.e., x((y1+y2/2))+y((x1+x2/2)) = 2((x1+x2/2))((y1+y2/2)) ⇒ x(y1+y2)+y(x1+x2)(y1+y2) ⇒ (x/x1+x2)+(y/y1+y2) = 1

Q. The equation of the chord joining two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ on the rectangular hyperbola $x y = c^{2}$ is

$\frac{x}{x _{1} + x _{2}} + \frac{y}{y _{1} + y _{2}} = 1$

$\frac{x}{x _{1} - x _{2}} + \frac{y}{y _{1} - y _{2}} = 1$

$\frac{x}{y _{1} + y _{2}} + \frac{y}{x _{1} + x _{2}} = 1$

$\frac{x}{y _{1} - y _{2}} + \frac{y}{x _{1} - x _{2}} = 1$

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

x1​+x2​x​+y1​+y2​y​=1

x1​−x2​x​+y1​−y2​y​=1

y1​+y2​x​+x1​+x2​y​=1

y1​−y2​x​+x1​−x2​y​=1

The mid -point of the chord is (2x1​+x2​​,2y1​+y2​​) The equation of the chord in terms if its mid-point is T=S1​, i.e., x(2y1​+y2​​)+y(2x1​+x2​​) =2(2x1​+x2​​)(2y1​+y2​​) ⇒x(y1​+y2​)+y(x1​+x2​)(y1​+y2​) ⇒x1​+x2​x​+y1​+y2​y​=1

Q. The equation of the chord joining two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ on the rectangular hyperbola $x y = c^{2}$ is

$\frac{x}{x _{1} + x _{2}} + \frac{y}{y _{1} + y _{2}} = 1$

$\frac{x}{x _{1} - x _{2}} + \frac{y}{y _{1} - y _{2}} = 1$

$\frac{x}{y _{1} + y _{2}} + \frac{y}{x _{1} + x _{2}} = 1$

$\frac{x}{y _{1} - y _{2}} + \frac{y}{x _{1} - x _{2}} = 1$