Expert: Paul Klarreich - 2/15/2011

Question
Show that the line (x-h)cos(alpha)+(y-k)sin(alpha)=a touches the circle (x-h)^2+(y-k)^2=a^2 for all (alpha)and find the coordinates of the point of contact.

Answer
Questioner:   Dhananjay
Category:  Advanced Math
Private:  No
 
Subject:  Maths-Circle
Question:  Show that the line (x-h)cos(alpha)+(y-k)sin(alpha)=a touches the circle (x-h)^2+(y-k)^2=a^2 for all (alpha)and find the coordinates of the point of contact.
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WLOG, assume h = k = 0  [Explanation on request.]

alPha will be written as 'p', to save typing.

x cos(p)+ y sin(p) = a

The slope of that line is - cot p.

Now take the line from the origin (and center of circle) which is perpendicular to that line --  its slope is the negative reciprocal, which is

m = tan p.

and whose equation is

y = x tan p

Where do these lines meet?  solve:


x cos(p)+ y sin(p)=a

y = x tan p

x cos(p)+ x tan p sin(p)=a

x cos(p)+ x sin^2(p)/cos p = a

x cos^2(p)+ x sin^2(p) = a cos p

x(cos^2(p)+  sin^2(p)) = a cos p

x(          1        ) = a cos p

x = a cos p

y = x tan p

y = a cos p tan p

y = a sin p

Now you can easily show that this is on the circle.