Question

Find the midpoint of the chord formed by the line
x+y+4=0 and the hyperbola y=1/x

Answer 1

Could I help you?

Answer 2

yeah

Answer 3

so What do you think?
Any ideas?

Answer 4

Did you not try to plot these two function and notice the relationship between them graphically?

Answer 5

Like that form example?
https://www.desmos.com/calculator/d8gcygp9qd

Answer 6

x+y+4=0
y=1/x
\[x \neq 0\]
\[x+\frac{ 1 }{ x }+4=0,x^2+4x+1=0\]
\[x=\frac{ -4\pm \sqrt{4^2-4*1*1} }{ 2*1 }=\frac{ -4 \pm \sqrt{12} }{ 2 }=-2 \pm \sqrt{3}\]
therefore two points of intersection are \[x1=-2+\sqrt{3},x2=-2-\sqrt{3}\]
mid point of x1 and x2 is
\[x=\frac{( -2+\sqrt{3})+(-2-\sqrt{3}) }{ 2 }=-2\]
x+y+4=0
\[when\ x=-2+\sqrt{3}\]
\[-2+\sqrt{3}+y+4=0,y=-2-\sqrt{3}\]
when \[x=-2-\sqrt{3},-2-\sqrt{3}+y+4=0,y=-2+\sqrt{3}\]
point of intersections are
\[\left( -2+\sqrt{3},-2-\sqrt{3} \right) \ and \ \left( -2-\sqrt{3},-2+\sqrt{3} \right)\]
Let (x,y) be the mid point of these two points.
then \[x=\frac{ x1+x2 }{ 2 },y=\frac{ y1+y2 }{ 2 }\]
x co-ordinate of mid point i have calculated.
similarly find y

Answer 7

Thank you for good explanation, @sshayer

Really really nice....