Question

well im doing conic sections (circles, ellipses etc.) and i need help solving this problem please i have a test nxt week n i dont feel ready

Answer 1

im drawing the porblem now :/

Answer 2

\[(x+6)^2+(y+4)^2=25 \] i need to know how to find a line thats tangent to the circle and intersects at the point (-9,-8)

Answer 3

Here's the process I'd take:
1. Pick a generic point \((x_0,y_0)\) that is on the circle, and find the equation of its tangent line. It'll be of the form \(y-y_0=m(x-x_0)\).
2. Plug in the given point for the value of \((-9,-8\). So, you'll have something like \(-8-y_0=m(-9-x_0)\).
3. Find a value for \((x_0,y_0)\) that satisfies both the circle equation as well as the equation I mentioned in step 2.

Answer 4

I'm not sure if this is how you're supposed to do it, but the way I'd do it is to first of all find the distance between (-9, -8) and the center of the circle.
Center: (-6, -4)
Radius: 5
Distance from center to (-9, -8): 5
THis problem is curious because (-9, -8) is on the circle. That means you have to find the slope of these two points.
Slope fo line from center to (-9, -8): 4/3
Now find the slope of a line perpendicular because a point of tangency creates a right angle.
Slope of tangent line: -3/4
y = -3/4x + b
-8 = -3/4(-9) + b
-8 = 27/4 + b
b = -59/4
y = -3/4x - 59/4 or y = -0.75x - 14.75

Answer 5

a line that is tangent to a circle at a point (a, b) will always be perpendicular with the radius draw to the point (a, b).

so find the slope of the radius. the slope mp is the negative reciprocal of the slope of the radius. since the point (a, b) is on the tangent line, you have the equation of the tangent line in point slope form:

y - b = mp*(x - a)

Answer 6

THANKS TO EVERYONE FOR HELPING ME :) I THINK I GOT IT NOW