Circle Geom: Show Line Tangent & Find Point of Contact

[Andy21]

Homework Statement

Show that the line 2x+3y=27 is a tangent to the circle with centre (4,2) and radius sqrt of 13. Find the co-ordinates of the point of contact. (Without a calculator)

Homework Equations

The Attempt at a Solution

I have worked out that the equation of the circle is (x-4)^2 + (y-2)^2=13.
I have tryed to substitute in values of x and y from the equation of the tangent to try and work out the discriminant but I haven't been able to create an equation with just either Xs or Ys in that is easy to work out the disccriminant of without using a calculator. Please show me the method for both parts of the question.

 

[HallsofIvy]

Since you are given the equation 2x+ 3y= 27, which is the same as y= 9- (2/3)x, at any point of intersection we must have [itex](x- 4)^2+ (9- (2/3)x- 2)^2= (x- 4)^2+ (7- (2/3)x)^2= 13. Multiply that out and you have a quadratic equation for x.

Now, there are three possibilities for a line and a circle:
1) they do not intersect at all.
2) they intersect in two different points.
3) they touch at one point.

and those correspond to the three possiblities for solutions of a quadratic equation:
1) there are two complex roots.
2) there are two real roots.
3) there is a single root.

In both situations, (3) is the case for a tangent line. Use the "discriminant" to show that this quadratic equation has a single solution.