What is the product of length of perpendicular drawn from any point on the hyperbola to its asymtotes. ..given only a and b...

Dear Student,

For the hyperbola x²/a² – y²/b² = 1 ,

Te two asymptotes are given by x/a + y/b = 0 or x/a- y/b=0. We can rewrite these as bx + ay = 0 and bx – ay = 0 respectively.

Now the length of perpendicular for any point P(x,y) on to these asymptotees are given by (bx+ay)/(sqrt(a²+b²)) and (bx-ay)/(sqrt(a²+b²)) respectively.

And thus the product of these lengths can be simplied easily to (b²x² – a²y²)/(a²+b²)

Thanks.