Does anyone know how to do this math problem??

Question

A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write the equation of a line tangent to the circle whose equation is x^2 + y^2 = 25. Can anyone help me with this? I know you have to use the slope formula some how but I'm not sure how. If you can help me, please show the steps. Thank you so much...

Yeah I forgot a part it connects at the point (3,-4)...sorry

Answer 1

The equation of the tangent line depends upon which point the line touches the circle. Didn't they give you a particular point to work the problem from?

For example, one point where a tangent might touch would be (0,5) and at that point, the equation of the tangent line would be y=5. another point where a tangent line might touch is (0,-5), where the tangent line equation would be y=-5 .

Answer 2

Well the equation X^2 + Y^2 = 25 is an equation of a circle centered around the origin with radius 5. Now since there are infinite positions that the radius could be (horizontal, vertical, diagonal, etc), then there are an infinite amount of possible tangent lines. Let's say, for example, the radius were going horizontal. So imagine an invisible segment going along the x-axis from (0,0) (the center of the circle) right to (5,0) (the right side of the circle). At that point (5,0), draw a line that is perpendicular (in this case, going vertically). The equation of this line would be x=5. The same could be the case for x=-5 if you decided to go the other way. Now let's say the radius was vertical, like for example going from (0,0) to (0,5). Then at the point (0,5) there would be a horizontal line (horizontal is perpendicular to vertical) and that equation is y=5. You can go on this way chosing different positions for the radius. As a general rule, to find the slope of a line that is perpendicular to another one, you find it's opposite inverse.

Answer 3

x^2 + y^2 = 25 is a circle of radius 5 with its center at (0,0). There are four tangents that are easy to come up with, those being x=5, x=-5, y=5, y=-5. These are tangent because they only touch the circle at the axis.

As for coming up with others that aren't at right angles, I'm not sure.

Answer 4

some lines include y=5, y=-5, x=5, x=-5

Answer 5

There are infinite number of lines tangent to that circle.
If we assume (x0,y0) is a point on the circle we can write the line:
y=m'(x-x0)+y0
m'=-1/m
m=y0/x0
==>m'=-x0/y0
==>y=-x0/y0(x-x0)+y0
if you choose 1 x0 you can have 2 y for that point.
y=+-sqrt(25-x0^2)
==>
y=-[x0/sqrt(25-x0^2)](x-x0)
+sqrt(25-x0^2)
and
y=[x0/sqrt(25-x0^2)](x-x0)
-sqrt(25-x0^2)