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.

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) (Show Source): You can put this solution on YOUR website!
The shortest distance would be a perpendicular line from the line,
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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,

Substitute into the hyperbola equation,

and
Then,

and
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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.

You have the slope and the point, find the line.

Now find the intersection point of the two lines to find the other point to use to calculate minimum distance.

Multiply both side by .

Then,

So now use the distance formula,

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You can also calculate the other point that intersects with (6,-3) and then calculate the distance.

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