Thnx Reiny & Steve 4 d help rendered in per the past questions. Plse help in this, If the line through the point y=2x+c touches the hyperbola 4(x)^2-6(y)^2=24. Show that c=+-2sqrt5.

Question

Thnx Reiny & Steve 4 d help rendered in per the past questions. Plse help in this,                                 If the line through the point y=2x+c touches the hyperbola 4(x)^2-6(y)^2=24. Show that c=+-2sqrt5.

Steve · Answer

4x^2 - 6y^2 = 24
8x - 12y y' = 0
y' = 2x/3y

So, at any point (x,y) on the hyperbola, the slope is 2x/3y.

The line 2x+c has slope 2.

So, where do we have 2x/3y = 2?
x = 3y

4x^2-6y^2=24
4*9y^2-6y^2=24
30y^2=24
y^2=4/5
y=2/√5

Things are lookin' good, with that √5 there. See where that gets you. Find x, then plug that in to get c.