Question

find the values of b for which the line 5x + by = 169 is a tangent to x^2 + y^2 =169...
please help

Answer 1

well, the b is part of slope for the line, and a derivative is the slope of an equation .... can you do a derivative?

Answer 2

i don't think so

Answer 3

hmm, what methods have you been working with then? circles dont need calculus, but it does tend to be useful

Answer 4

im sure calculus would make it easier by far; are you in calculus and just learning derivatives by chance?

Answer 5

well this is under the geometry chapter

Answer 6

hmm

well, the y intercept of the line would need to be greater that 14 to at least have shot at being tangent to the circle

169/b > 14; 169/14 > b ... 14 > b

Answer 7

the slope of the line itself is -5/b; which means that the slope would be -5/14 or smaller

Answer 8

well the way we have been doing it is we get the radius and then use the distance formula ?

Answer 9

that does sound like something more doable to me :)

radius is sqrt(169) = 14 and the slope of the radius would have to be b/5 to be perp to the line

Answer 10

is the radius not 13 ?

Answer 11

lol .... fine, if you wanna be "correct" and all.

12: 144
13: 169 yeah it is ... i tend to get it mixed around with the 14
14: 196

Answer 12

ah its fine i did in the calculator haha

Answer 13

line of the radius would have to be:

y = b/5 x

line of the tangent is given as:

y = -5/b x + 169/b

any way to compare the 2?

Answer 14

divide by b ?

Answer 15

y' = -n/m for any point (n,.m) on the circle

y = -n/m (x-n) + m

would be the tangent line of the circle and this has to equal the given line, with the same point (n.m)

5n +bm = 169


\[y = -\frac{n}{m}x+\frac{n^2+m^2}{m} \]
5n +bm = 169

comparing intercepts; i would say that 5^2+b^2 = 169

b^2 = 144

b = -12 , 12

lets see if the wolf likes it :)

http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%3D169%2C+5x%2B12y%3D169%2C+5x-12y%3D169

Answer 16

not the way your used to seeing it, but it was the only way that made sense to me :)

Answer 17

ok it ok thanks well the answer in the back of the book is plus or minus 12

Answer 18

so you are right anyway

Answer 19

yay!! ... i got no idea how id try to approach it using the distance and radius stuff. Either too long out of geometry or never really learnt it to begin with

Answer 20

haha thanks anyway its just my teacher she like to make thing complicated

Answer 21

good luck ;)

Answer 22

thanks