Problem

d=|ax0+by0+c|a2+b2

Answer

Expert–verified
Hide Steps
Answer

d=a2(x02a2+b2)+2ab(x0y0a2+b2)+b2(y02a2+b2)+2ca(x0a2+b2)+2cb(y0a2+b2)+c2(1a2+b2)

Steps

Step 1 :Let x=a2+b2. Then x2=a2+b2, so d=|ax0+by0+c|x

Step 2 :We can rewrite the equation as |ax0+by0+c|=dx

Step 3 :Square both sides to get rid of the absolute value: (ax0+by0+c)2=d2x2

Step 4 :Expand the equation: a2x02+2ax0by0+b2y02+2cx0a+2cy0b+c2=d2(a2+b2)

Step 5 :Substitute x2 back into the equation: a2x02+2ax0by0+b2y02+2cx0a+2cy0b+c2=d2x2

Step 6 :Divide both sides by x2: a2x02+2ax0by0+b2y02+2cx0a+2cy0b+c2x2=d2

Step 7 :Simplify the equation: a2x02x2+2ax0by0x2+b2y02x2+2cx0ax2+2cy0bx2+c2x2=d2

Step 8 :Cancel out the common terms: a2(x02x2)+2ab(x0y0x2)+b2(y02x2)+2ca(x0x2)+2cb(y0x2)+c2(1x2)=d2

Step 9 :Simplify further: a2(x02a2+b2)+2ab(x0y0a2+b2)+b2(y02a2+b2)+2ca(x0a2+b2)+2cb(y0a2+b2)+c2(1a2+b2)=d2

Step 10 :Now, we can see that the equation is in the form of d2, so we can take the square root of both sides to get the final answer: d=a2(x02a2+b2)+2ab(x0y0a2+b2)+b2(y02a2+b2)+2ca(x0a2+b2)+2cb(y0a2+b2)+c2(1a2+b2)

Step 11 :d=a2(x02a2+b2)+2ab(x0y0a2+b2)+b2(y02a2+b2)+2ca(x0a2+b2)+2cb(y0a2+b2)+c2(1a2+b2)

link_gpt