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 x2+y2=25.



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


Substituting this for y in the equation of the circle,

x2 +(25-4x3)2 =25

Squaring out the bracket and multiplying by nine,

9x2 +(25-4x)2 =225

9x2 +625-200x+16x2 =225

25x2 -200x+400 =0

x2 -8x+16=0

Using the determinant on this quadratic,

b2-4ac =82-4×1×16=0

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


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