Final Answer:
The equation y=mx+c can be reduced to the form x cos a+y sin a=p, where a is the angle whose tangent is m, and p is the distance of the line from the origin. Additionally, 1/p²=m²/c²+1/c².
Step-by-step explanation:
To reduce the equation y=mx+c in the form of x cos a+y sin a=p, we start by expressing m as tan a. Therefore, m=tan a. Then, we rewrite the given equation as y=tan ax+c. We can then express this in terms of sine and cosine using the identity tan a=sin a/cos a.
This gives us y=(sin a/cos a)x+c, which simplifies to ycos a=xsin a+ccos a. Rearranging terms gives us xcos a+y*sin a=p, where p=c/sin a. To prove 1/p²=m²/c²+1/c², we start with p=c/sin a and square both sides to get p²=c²/sin²a. Using the identity sin²a=1-cos²a and substituting for sin²a, we get p²=c²/(1-cos²a).
Then, we use the identity 1-cos²a=sin²a to obtain p²=c²/sin²a. Finally, substituting m=tan a into p²=c²/sin²a gives us 1/p²=m²/c²+1/c².