# Is it a tangent to the circle?

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

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### 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,

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

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

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

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

Using the determinant on this quadratic,

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

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

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