General equation of a line tangent to a circle

Consider x^2+y^2=r^2 and y=mx + b. find the value(s) of b in terms of r and m so that the graphs of the two given equations will intersect at one and only one point.

I've thought of b = +Sqrt(r^2+m^2) and b = -Sqrt(r^2+m^2)
but it seems that it only works if r=1.

Can anybody help?

Consider x^2+y^2=r^2 and y=mx + b. find the value(s) of b in terms of r and m so that the graphs of the two given equations will intersect at one and only one point
Sol: x^2+y^2 = r^2.................(i)
and y = mx +b...................(ii)
To find the point of intersection of (i) and (ii)
x^2 +(mx+b)^2 =r^2
x^2 +m^2x^2 +b^2 +2mbx = r^2
x^2(1+m^2) + 2mbx + (b^2 -r^2) = 0

Since both roots are equal when st line touch circle
Discriminant = (2mb)^2 - 4(1+m^2)(b^2 -r^2) = 0
4m^2b^2 = 4{b^2 -r^2 +m^2b^2 -m^2r^2}
m^2b^2 = {b^2 -r^2 +m^2b^2 -m^2r^2}
0 = {b^2 -r^2 -m^2r^2}
b^2 = r^2 +m^2r^2
= r^2(1+m^2)
b = (+/-) r*sqrt(1+m^2).............Ans

Note that the slope of a line tangent to the circle is the derivative of the circle at the point in question.

Find y’: d(x² + y²)/dx = dr²/dx means 2x + 2yy’ = 0

∴ y- = -x/y = m {the x and y are the coordinates of the point in question}

∴ the equation of the tangent line is y = (-x/y)x + b,

b = y + x²/y = y + (r² ‒ y²)/y = y + r²/y – y = r²/y

This tangent line is y = (-x/y)x + r²/y

Example: find the line tangent to x² + y² = 25 at (3, 4)
From above the tanent line’s equation is y = -3x/4 + 25/4 = (25 ‒ 3x)/4

You try x² + y² = 36 at (1, √35).

ProfRay

x^2 + y^2 = r^2
2x * dx + 2y * dy = 0
x * dx + y * dy = 0
x * dx = -y * dy
dy/dx = -x / y
dy/dx = -x / +/- sqrt(r^2 - x^2)
dy/dx = +/- x / sqrt(r^2 - x^2)

This gives you the slope of the tangent line at point (x , y)

Supposing that x = a, then y = +/- sqrt(r^2 - a^2)

m = +/- a / sqrt(r^2 - a^2)

y = mx + b
+/- sqrt(r^2 - a^2) = +/- (a / sqrt(r^2 - a^2)) * a + b
+/- sqrt(r^2 - a^2) +/- (a^2 / sqrt(r^2 - a^2)) = b
+/- ((r^2 - a^2 +/- a^2) / sqrt(r^2 - a^2)) = b
+/- r^2 / sqrt(r^2 - a^2)) = b

y = +/- (1 / sqrt(r^2 - a^2)) * (x + r^2) is the general formula for a tangent line