A Circle centered at (2,3) has a radius of 5. What is the distance of point (9,-1) to the tangent line of the circle at x=6?

Question

A Circle centered at (2,3) has a radius of 5. What is the distance of point (9,-1) to the tangent line of the circle at x=6? The shortest distance

Answer 1

The point of tangency is (6, 0). This can be determined by solving the equation of the circle using x = 6.

The tangent is perpendicular to the radius at the point of tangency. The slope of the radius to the point (6, 0) is -3/4.

The slope of the tangent line is 4/3.

 

Using the slope-point form

 

4/3 = (y - 0) / (x - 6)

4(x - 6) = 3y

4x - 24 = 3y

4x/3 - 8 = y                 This is the equation of the tangent line.

 

The equation of the line trough (9, -1) and perpendicular to the tangent is

 

-3/4 = (y + 1) / (x - 9)

-3(x - 9) = 4( y + 1)

-3x + 27 = 4y + 4

-3x + 23 = 4y

-3x/4 + 23/4 = y

 

Using the equation of the tangent and the equation of the line passing through (9, -1), we find the point of intersection

 

4x/3 - 8 = -3x/4 + 23/4                  Common denominator is 12

16x - 96 = -9x + 69

25x = 165

x = 6.6                                            Substituting this value into the equation of the tangent line

 

y = 4(6.6)/3 - 8

y = 8.8 - 8

y = 0.8                                            The point of intersection is (6.6, 0.8)

 

Use this point of intersection and the point (9, -1) to calculate the distance using the distance formula.

 

 

Answer 2

Osiris,

 

I have a different take on the problem than the previous responses...and I think you're right that this is more complex than just sketching a drawing.

 

- We need the distance between a point and a line.

- We're given the point (9,-1).  At least that part's done!

- If we knew the equation of the line, then we could figure out the distance between (9,-1) and that line.  (There's actually an equation for the distance between a point and a line, but even without the equation, it can be solved with a bunch of steps.)

 

So, what we need is the equation of the line.  In this problem, we're looking for the line that is tangent to the circle at x=6. 

 

- To write the equation of a line, we'd need a point on the line, and its slope.

- Well, the x coordinate of the point is 6, and that point is on the circle described in the problem.  So, if we knew the equation of that circle, I think we could figure out the y coordinate that goes with x=6?

- For the slope of the tangent line....don't we know something about a tangent to a circle and the radius of the circle that is drawn to the same point?  Draw a radius of the circle to where the tangent line meets the circle...and there's a relationship between those two lines, which should help with the slope.

 

If we put all of that together, we should be able to turn this into a problem where you have a point and the equation of a line, and we need to figure out the distance between them.

 

I've left some gaps for you to fill in, but does any of that help get you started?

 

Answer 3

First draw the circle on a coordinate system.  Then plot the point (9,1).  Draw a line from that point to the point that is tangent at x=6.  After that, use the distance formula to get your answer:

 

d = sqrt[(x₂ - x₁)² + (y₂ - y₁)²]