SOLUTION: On the hyperbola x^2/24-y^2/18=1 find the point M1 nearest to the line 3x+2y+1=0, and compute the distance d from M1 to the line.

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: On the hyperbola x^2/24-y^2/18=1 find the point M1 nearest to the line 3x+2y+1=0, and compute the distance d from M1 to the line.      Log On


   

Question 913268: On the hyperbola x^2/24-y^2/18=1 find the point M1 nearest to the line 3x+2y+1=0, and compute the distance d from M1 to the line.
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The shortest distance would be a perpendicular line from the line, 3x%2B2y%2B1=0
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2y=-3x-1
y=-%283%2F2%29x-1%2F2
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A point on the hyperbola that had the same slope would be at the minimum distance.
Find the slope of the hyperbola by differentiating,
x%5E2%2F24-y%5E2%2F18=1
%281%2F24%29%282xdx%29-%281%2F18%29%282ydy%29=0
%28xdx%29%2F12=%28ydy%29%2F9
dy%2Fdx=%283x%29%2F%284y%29=-3%2F2
6x=-12y
x=-2y
Substitute into the hyperbola equation,
%28-2y%29%5E2%2F24-y%5E2%2F18=1
y%5E2%2F6-y%5E2%2F18=1
%283%2F18%29y%5E2-%281%2F18%29y%5E2=1
%282%2F18%29y%5E2=1
y%5E2=9
y=-3 and y=3
Then,
x%5E2%2F24-9%2F18=1
x%5E2%2F24=3%2F2
x%5E2=36
x=6 and x=-6
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So the two points are (-6,3) and (6,-3).
Now the perpendicular line that would be the shortest distance to the original line has a slope that is the negative reciprocal of the original line.
m=2%2F3
You have the slope and the point, find the line.
y-3=%282%2F3%29%28x-%28-6%29%29
y-3=%282%2F3%29%28x%2B6%29
y-3=%282%2F3%29x%2B4
y=%282%2F3%29x%2B7

Now find the intersection point of the two lines to find the other point to use to calculate minimum distance.
y=-%283%2F2%29x-1%2F2
y=%282%2F3%29x%2B7
-%283%2F2%29x-1%2F2=%282%2F3%29x%2B7
Multiply both side by 6.
-9x-3=4x%2B42
-13x=45
highlight%28x=-45%2F13%29
Then,
y=%282%2F3%29%28-45%2F13%29%2B7
y=-%2830%2F13%29%2B91%2F13
highlight%28y=61%2F13%29
So now use the distance formula,
D%5E2=%28-6-%28-45%2F13%29%29%5E2%2B%283-61%2F13%29%5E2
D%5E2=%28-78%2F13%2B45%2F13%29%5E2%2B%2839%2F13-61%2F13%29%5E2
D%5E2=%28-33%2F13%29%5E2%2B%2822%2F13%29%5E2
D%5E2=%281089%2F169%29%2B%28484%2F169%29
D%5E2=1573%2F169
D=sqrt%281573%29%2F13
highlight%28D=%2811%2F13%29sqrt%2813%29%29
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You can also calculate the other point that intersects with (6,-3) and then calculate the distance.