Question

(STILL STUCK!) The following equation represents a circle. Find the gradient of the tangent at the given point (i) by finding the coordinates of the centre, and (ii) by differentiating the implicit equation.

(a) x^2 +y^2 =25

Answer 1

dy/dx=-(x/y)

Answer 2

So, how?

Answer 3

so we're finding dy/dx right?

Answer 4

that'll do it
you can even solve for y and then differentiate if you like. get the same answer

Answer 5

if the gradient of the line joining centre and the point is m, then the gradient of the tangent will be -1/m

Answer 6

and the differentiation method, anonymous... is right ^^

Answer 7

intermediate step is
\[x^2+y^2=1\]
\[2x+2yy'=0\] then you get
\[y'=-\frac{x}{y}\]

Answer 8

Still a bit lost.

Answer 9

or you can start
\[x^2+y^2=1\]
\[y=\pm\sqrt{1-x^2}\]
\[y'=\pm\frac{x}{\sqrt{1-x^2}}\] same thing

Answer 10

Why equal to 1? I thought 25?

Answer 11

typo, although it cannot make much of a difference

Answer 12

my cirlce has radius 1 and yours has radius 5

Answer 13

using partial derivatives, implicit differentiation is less laborous... all you have to do is set get this to a form F(x,y)=0.
In this case
x^2+y^2-25=0
and we know that
\[dy/dx=-F_x/F_y\]
where \[F_x\] is the partial derivative of F(x,y) with respect to x and \[F_y\] is the partial derivative with respect to y

Answer 14

so would it be x/ sqrt {25 - x^2}?

Answer 15

@satellite73

Answer 16

yes plus or minus

Answer 17

i already did, it is the derivative

Answer 18

you have a choice starting with
\[x^2+y^2=25\]a) solve for y or
b) pretend you have solved for y

Answer 19

if you solve for y you get
\[y=\pm\sqrt{25-x^2}\] and then take the derivative of that
you get
\[y'=\pm\frac{x}{\sqrt{25-x^2}}\]

Answer 20

or you pretend
\[y=f(x)\] and take the derivative of
\[x^2+f^2(x)=25\] to get
\[2x+2f(x)f'(x)=0\] solving for
\[f'(x)\] you get
\[f'(x)=-\frac{x}{f(x)}\]
more easily written as
\[x^2+y^2=25\]
\[2x+2yy'=0\]
\[y'=-\frac{x}{y}\]

Answer 21

So how would you get the values in the first equation once plugging in (-3 and 4)?

Answer 22

Oh, Ok, -3/4 right?

Answer 23

But the answer says 3/4?