Circle Geom: Show Line Tangent & Find Point of Contact

In summary, to show that the line 2x+3y=27 is a tangent to the circle with centre (4,2) and radius sqrt of 13, we first find the equation of the circle, which is (x-4)^2 + (y-2)^2=13. Then, by substituting in values of x and y from the equation of the tangent, we can create a quadratic equation and use the discriminant to determine the number of solutions. Since we are looking for a tangent line, we need to show that the quadratic equation has a single solution, indicating that the line and circle intersect at only one point. This will also give us the coordinates of the point of contact.
  • #1
Andy21
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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.
 
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  • #2
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.
 

Related to Circle Geom: Show Line Tangent & Find Point of Contact

1. What is a tangent line in circle geometry?

A tangent line is a line that intersects a circle at exactly one point, called the point of tangency. This point is where the line touches the circle and is perpendicular to the radius at that point.

2. How do you show a line is tangent to a circle?

To show that a line is tangent to a circle, you must prove that it intersects the circle at exactly one point and is perpendicular to the radius at that point. This can be done using the tangent theorem or by showing that the line is perpendicular to the radius at the point of tangency.

3. What is the process for finding the point of contact between a line and a circle?

The point of contact, or point of tangency, between a line and a circle can be found by setting the equation of the line equal to the equation of the circle and solving for the x and y coordinates of the point of contact. This can also be done graphically by finding the intersection point between the line and the circle.

4. Can a line be tangent to a circle at more than one point?

No, a line can only be tangent to a circle at one point. This is because a tangent line must intersect the circle at exactly one point and be perpendicular to the radius at that point.

5. What is the significance of finding the point of contact between a line and a circle?

Finding the point of contact between a line and a circle is important in many applications of geometry and mathematics. It allows us to determine the angle of intersection between the line and the circle, as well as the length of the tangent line segment. This information can be used in various calculations and proofs.

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