Posted by **Adrianna** on Wednesday, April 6, 2011 at 10:48pm.

point on the hyperbola 9x^2−6y^2=10 closest to the point (0, 7).

y coordinate of each point= ?

positive x coordinate= ?

negative x coordinate= ?

- CALC -
**Reiny**, Wednesday, April 6, 2011 at 11:56pm
let the point be P(a,b)

slope of line form P to (0,7) = (b-7)/a

differentiate:

18x - 12y dy/dx = 0

dy/dx = 18x/(12y) = 3x/(2y)

at P, dy/dx = 3a/(2b)

so the slope of the tangent at P is 3a/(2b)

by basic geometry the slope of that tangent and the slope of the line to (0,7) to P must be negative reciprocals of each other.

so (b-7)/a = -2b/(3a)

-2ab = 3ab - 21a

5ab = 21a

b = 21/5

sub into 9a^2 - 6b^2 = 10

9a^2 - 6(441/25) = 10

a = ± 4√181/15

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