Finding tangents

Question

Show that 3x + 4y + 25 = 0 is a tangent to the circle x^2 + y^2 = 25

Answer 1

Use Substitution to find a common point.

Solve for y to get

y = (1/4) (-3x + 25 )

y^2 = (1/16) (-3x+25)^2

Substitute in to x^2+y^2=25

With careful arithmetic you should arrive at

x^2-6x+25=0  => x=3, y=4

This point is also on the circle.

Next you must show the radius to that point

is perpendicular to the line.

You do this by showing the product of the slopes

of the radius and the line = -1

Radius line passes thru (0,0) and (3,4) => m=4/3

Slope 3x+4y=25 => m=-3/4

qed