How to find the equation for a tangent line with a given y intercept.

Question

So, I have an equation for a circle, $y^2+x^2=3959^2$ and I have the y intercept for a tangent line $y=mx+3965$.
I need to find the equation for the tangent line (the slope)
I have no idea where the first step is, any help would be appreciated.

Answer 1

What I did to solve this, was finding the distance between $0,3965$ and the tangent point, by using Pythagorean theorem,
$$3965^2=3959^2+b^2$$ The distance I found was,
$$2\sqrt(11886)$$ Now I made this the solution of the distance formula,
$$((0-x)^2+(3965-f(x)))^{\frac12}=2\sqrt(11886)$$
I found that the point at which it is tangent is at, $$x=\frac{7918\sqrt11886}{3968}$$
I plugged this value into the derivative of the equation for the circle,
$$-\frac{x}{15673681-x^2}$$
And the solution for the slope of the tangent line is, $$-\frac{2 \sqrt(11886}{3959}$$
EDIT
If anyone is viewing this becuase they want to know the answer to the question stated above, I made a little formula to find the slope of a circle with a given radius and a given y-intercept for the tangent line.
$$m=\pm\frac{-\sqrt{\left|r^2-b^2\right|}}{r}$$

Answer 2

HINT

Consider the system

• $y^2+x^2=3959^2$
• $y=mx+3965$

substitute $y$ in the equation of the circle and, for the quadratic equation obtained, impose that the discriminant is equal to $0$, that is the tangency condition.

Answer 3

If point of tangent is $(x_1,y_1)$ where $x_1 \ne 0$, then $$y_1^2+x_1^2=3959^2 \space \space \space (1)$$ $$y_1=mx_1+3965\space \space \space (2)$$ and applying tangent-radius relationship, $$\frac{y_1}{x_1}\times m=-1$$ Hence, $$ y_1=-\frac{x_1}{m} \space \space \space (3)$$

Substituting $y_1$ in equation (2) gives, $$-\frac{x_1}{m}=mx_1+3965$$ $$x_1=-\frac{3965}{(m^2+1)} \space \space \space (4)$$

When substitute the value of $x_1$ in equation (4) in equation (3) gives,

$$ y_1=\frac{3965}{(m^2+1)}\cdot \frac{1}{m} \space \space \space (5)$$

Now you have both $x_1$ and $y_1$ in $m$ in equations (4) & (5), so substituting those values in equation (1) give you the value of $m$.