1.(a) A point moves on hyperbola 3x^2 -y^2 = 23 so that its y-coordinate is increasing at a constant rate of 4 units per second. How fast is the x coordinate changing @ x=4?
(b) For what values of k will the line 2x + 9y + k = 0 be normal to the hyperbola 3x^2 -y^2 = 23?
(b) For what values of k will the line 2x + 9y + k = 0 be normal to the hyperbola 3x^2 -y^2 = 23?
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(a)
3x^2 -y^2 = 23
plug x=4 ==> 48-y^2=23==> y^2=25 ==> y= +/-5
Take derivatie
6xdx - 2ydy = 0 ==> dx = (y/3x)dy
at x= 4
dx = +/- (5/12)dy
dy = 4 ==> dx = +/- (5/3)
b)
3x^2-y^2 = 23
dy/dx = 3x/y
slope of normal line = -y/3x
slope of 2x+9y+k = -2/9
==> -y/3x = -2/9 ==> y/x = 2/3 or 2x=3y (at intersection point)
plug 2x=3y in the equation for hyperbola:
3x^2 - (4/9)x^2 = 23 ==> x^2 = 9 ==> x= +/-3, and y = +/-2
plug (+/-3, +/-2) in the equation of the normal line to get k
k = -(2x+9y) = +/-6 +/-18 = +/-24
3x^2 -y^2 = 23
plug x=4 ==> 48-y^2=23==> y^2=25 ==> y= +/-5
Take derivatie
6xdx - 2ydy = 0 ==> dx = (y/3x)dy
at x= 4
dx = +/- (5/12)dy
dy = 4 ==> dx = +/- (5/3)
b)
3x^2-y^2 = 23
dy/dx = 3x/y
slope of normal line = -y/3x
slope of 2x+9y+k = -2/9
==> -y/3x = -2/9 ==> y/x = 2/3 or 2x=3y (at intersection point)
plug 2x=3y in the equation for hyperbola:
3x^2 - (4/9)x^2 = 23 ==> x^2 = 9 ==> x= +/-3, and y = +/-2
plug (+/-3, +/-2) in the equation of the normal line to get k
k = -(2x+9y) = +/-6 +/-18 = +/-24
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" C B " 's analysis is fine , except when x = 4 and dy / dt = + 4 then y ≡ 5 , not ± 5