Question

Write the equation of the line that is tangent to the circle (x – 2)^2 + (y – 4)^2 = 289, at the point (-15, 4).

Answer 1

Have you tried to solve this question?

Answer 2

Yes. I've only been stuck on 2 problems for 30 minutes. The one I was helped with and this one.

Answer 3

Well, here also, differentiation will be involved. Can you tell me, what will be the differentiation of (x-2)^2 w.r.t x ?

Answer 4

I'm scared to guess...

Answer 5

Not to guess... ! Actually, it is :
Differentiation of \(x^n\) with respect to x is \(nx^{n-1}\)

So, let us first take (x-2) as a function (f(x)) :

Can you now tell me what will be the differentiation of \((f(x))^2\) w.r.t x ? (Hint : Just think that f(x) is x and n = 2 , put these values in the above formula.)

Answer 6

Not exactly.. Its not that simple. I think, you need to study differentiation for this,

Have a look at this link :

http://www.mathsisfun.com/calculus/derivatives-introduction.html

Answer 7

since (-15,4) lies on the circle (x – 2)^2 + (y – 4)^2 =289

Answer 8

now we know tangent is aline whicj is perpendicular to radius..

Answer 9

What matricked said :is actually proven through differentiating the given equation (as it gives you 17/0 that is infinite rise) and thus, tangent will be perpendicular line for the given y -coordinate.

Answer 10

centre of the above circle is (2,4) hence the line y = 4 passes through the centre (2,4) and the (-15,4) thus the required tangent line will be a line which is perpendicular to y=4 and passes through(-15,4) so the required line is x=-15

Answer 11

I really don't get this. I'm lost and confused. will that webpage really explain it all to me? I'm horrible at math & it sucks cause I have an exam on this stuff coming up soon :/

Answer 12

Did you try to understand matricked's method?

Answer 13

you have one point on the line, what else do you need to find to get the equation of the line ?

Answer 14

If not differentiation, this can be solved by similarity in triangles.

Answer 15

don't go for complications
no differentiation, no similar triangles

we have 2 points on a line, find slope
since radius is perpendicular to slope, we get the slope of tangent
since we have point on tangent, we get equation of tangent too

Answer 16

I'm barely learning all this stuff. Sorry. I really don't know most of it. I do know similarity of triangles. But I would like to learn in depth about differentiation. Math is my weakest subject. But thank you all for helping >.<

Answer 17

easy question :
find centre of this circle, can you ?
(x – 2)^2 + (y – 4)^2 = 289

Answer 18

Oh haha! Excellent! @vachave412 Try what @hartnn just said.
No differentiation is required here!
Nor is any similarity theorem needed to solve this;

Just plane ol' slopes and lines formulae. :D

Answer 19

(2,4) @hartnn ? idk...

Answer 20

*plain

Answer 21

2,4 is correct!

2nd easy question :
point (-15, 4) lies on this circle or not ?

Answer 22

yes........................

Answer 23

good
so as of now we have this :

Answer 24

i just joined those 2 points.

do you know how to get slope of line when we have 2 points on it ?

Answer 25

Formula :
The slope of the line through points (x1,y1) and (x2,y2) is given by :
\(\huge m=\frac{y_1-y_2}{x_1-x_2}\)
now,just put the values and find m.

Answer 26

* would be

Answer 27

there are only 2 points
2,4 as center and -15,4 as end-point on circle

so, to find the slope between them,
take
x1 = 2, y1 = 4
x2 =-15, y2 = 4

calculate m

Answer 28

ok i got this!!:)

Answer 29

0/17?....so the slope is.... 0 ?..... no? ok.. haha

Answer 30

yes! slope =0 is correct!

so, the radius segment, of which we found the slope as 0, is HORIZONTAL!

Answer 31

now a couple of very important theorems :
"Radius segment is perpendicular to Tangent"

Answer 32

since radius segment is HORIZONTAL,
tangent line will be VERTICAL

see if this makes sense?

Answer 33

Yayyyy! you're awesome thank you:) I love how you took the time to explain it. & everything makes sense:)

Answer 34

good, since tangent is vertical, and point -15,4 lies on it,
can you find its equation ?

Answer 35

hint :
equation of vertical line is of the form : x = c
where c is the x co-ordinate of any point on the line

Answer 36

(x+15)^2+(y−4)^2=289 ?

Answer 37

i was asking for the equation of tangent line.

since its Vertical, it has the equation in the form of x= c

since point (-15,4) lies on it, c = x co-ordinate = -15

so the equation just becomes
x = -15

thats it! got this ?

Answer 38

oooooh lol . yes >.< thank you. thanks for being so helpful & patient

Answer 39

welcome ^_^
review all the steps, ask if any doubt in any step.
make yourself very clear on this, so that you can solve such problems very easily on your exam :)

Answer 40

awww, will do!! you have a good night, thanks! :')

Answer 41

you have a great night with sweet dreams, good luck for the exam :)