Show that the line 2x - 3y + 36 = 0 is a tangent to the circle x^2 + y^2 - 4x + 6y - 104 = 0 state any theorem used without proof

Answer 1

With calculus, you can just take the derivative of the equation for the circle, plug in for the point where the circle and the line intersect, and show that the slope of the tangent line is the same as the slope of 2x-3y+36=0.

Without calculus, you will have to:

1. Find the point where the line is tangent to the circle (where they intersect). You can do this by solving for y with the line, substituting this into the circle, solve for x, then plug back in to get y. There should only be one point.
2. Find the slope of the line going through the center and the point from step 1.
3. Find the slope of 2x-3y+36=0, and show that this slope is perpendicular to the slope from step 2.

That's it! Step 1 shows the actual point the line is tangent to, and steps 2 and 3 show that the line is tangent by proving that it is perpendicular to the radius there.

Answer 2

I'm not sure about what theorem this is, but--for the line to be tangent---there must be only one solution for x and y. We have 2 equations, so we can solve using the normal methods.

I checked using an online graphing tool and found no common solution

here is the link:

https://www.desmos.com/calculator/hgrjtrz7im