Is it a tangent to the circle?

Show that the straight line with equation $4x+3y=25$ is a tangent to the circle with equation $x^2+y^2=25$.

-oOo-

Solution.

There are two ways in which a circle and a line might intersect.
They might also not intersect at all. A circle and a line might intersect once or twice.
A singular intersection can only occur if the line is a tangent, so for this problem we need only show that there is one intersection.
The equation for the line can be re-written as

$y=\frac{25-4x}{3}$

Substituting this for y in the equation of the circle,

$x^2+{\left(\frac{25-4x}{3}\right)}^2=25$

Squaring out the bracket and multiplying by nine,

$9x^2+{(25-4x)}^2=225$

$9x^2+625-200x+16x^2=225$

$25x^2-200x+400=0$

$x^2-8x+16=0$

Using the determinant on this quadratic,

$b^2-4ac=8^2-4\times 1\times \mathrm{16}=0$

This shows there is only one intersection, so the line is a tangent to the circle.

-oOo-