SOLUTION: explain why radius squared divided by y1 is the y-intercept in the equation of the line tangent to the circle x^2+y^2=r^2 like in this example: x^2+y^2=17 and the circle meets t

Click here to see ALL problems on Quadratic-relations-and-conic-sections

Question 294471: explain why radius squared divided by y1 is the y-intercept in the equation of the line tangent to the circle x^2+y^2=r^2
like in this example: x^2+y^2=17 and the circle meets the tangent at (1,4).What is the equation of the tangent line. The answer is y=(-1/4)x+(17/4)
Why does the (17/4) equal the radius squard by y1?
Found 2 solutions by scott8148, Edwin McCravy:
Answer by scott8148(6628) (Show Source):
You can put this solution on YOUR website!
it works, in this case, because the circle is centered at the origin

since the center is at the origin, the slope of a radius to any point (x,y) on the circle is y/x

a tangent is perpendicular to the radius at the point of tangency, so the slope of the tangent is -x/y

substituting, the equation for the tangent is ___ y = (-x/y)x + k

multiplying by y ___ y^2 = (-x)x + (ky) ___ y^2 = -x^2 + (ky)

adding x^2 ___ y^2 + x^2 = ky

dividing by y ___ (y^2 + x^2) / y = k ___ r^2 / y = k

Answer by Edwin McCravy(20060) (Show Source):
You can put this solution on YOUR website!

AB is tangent to circle O, whose equation is . OT is a radius (of length r) drawn to T(x₁,y₁), the point of tangency. Therefore OT is perpendicular to AB. OB has length b, the y-coordinate of the y-intercept of AB and B is the point (0,b). All triangles in the figure above are right triangles, and all of them are similar! This is easy to see if you realize that the pair of acute angles in each of them are equal in measure. In particular since triangles OTB and TDO are similar, Observing their lengths: Edwin