# CALC

posted by .

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 -

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

## Similar Questions

1. ### Pre-Algebra

The x- coordinate is 0 and the y-coordinate is negative. Is the point in quadrant 3?
2. ### math

A,B are two points on a given line L.L is the midpoint of the section AB. find the coordinate of point B if the coordinate of A=7 and the coordinate of L=4
3. ### math

A,B,C are points on the line L, the coordinate of B=-2, c is a point that divides AB in the ration 2:3.if the coordinate of C=4, find the coordinate of A
4. ### statistics

a point is randomly selected within the rectangle with vertices at (2,0), (2,2), (-4,2), and (4,0). What is the probability that the point's x-coordinate and y-coordinate are both positive?
5. ### Calculus-Applied Optimization Problem:

Find the point on the line 6x + 3y-3 =0 which is closest to the point (3,1). Note: Your answer should be a point in the xy-plane, and as such will be of the form (x-coordinate,y-coordinate)
6. ### Math

In a coordinate plane, Brittany labeled the coordinates for point m as (-31/2,2). Sade tells Brittany to reflect point m over the x axis to point r. What are the coordinate of point r?
7. ### calc

A particle moves along the curve below. y = sqrt(8 + x^3) As it reaches the point (2, 4), the y-coordinate is increasing at a rate of 5 cm/s. How fast is the x-coordinate of the point changing at that instant?
8. ### Calculus

A point is moving along the curve xy=12. When the point is at (4,3), the x-coordinate decreases at the rate of 2cm/sec. How fast is the y-coordinate changing at that point?
9. ### Math

1. How can you represent a loss of 24 dollars as an integer?
10. ### Math

1. How can you represent a loss of 24 dollars as an integer?

More Similar Questions