# Calculus w/ Analytical Geometry question!?

I have already learned methods of differentiations and integrations. However, it seems that i still do not have enough knowledge to be able to do this problem:

Q:

Find the coordinates of all points (x,y) on the graph of the equation x^2-xy+y^2 = 12 where the tangent is parallel to the y-axis.

My approach so far is to use implicit differentiation and plug in y=0, but i do not know what to do after this step.

Thank you if you can help me!

BY the way, the answer given is : (-4,-2) & (4,2)

### 2 Answers

- HemantLv 79 years agoBest Answer
... x² - xy + y² = 12 ..... (1)

∴ diff... w.r.t. x by Chain Rule,

... 2x - ( x.y' + y.1 ) + 2y.y' = 0

∴ ( 2y - x ) y' = y - 2x

∴ y' = ( y - 2x ) / ( 2y - x )

∴ slope of normal = -1/ y' = - ( 2y - x ) / ( y - 2x ) ....... (2)

Since tgt is parallel to Y-axis ∴ normal is parallel to X-axis.

∴ slope of normal = 0 ∴ from (2), ... 2y - x = 0 ∴ x = 2y

∴ from (1),

... (2y)² - (2y)y + y² = 12 ∴ 3y² = 12 ∴ y² = 4 ∴ y = ± 2

∴ x = 2y = 2 ( ± 2 ) = ± 4

∴ x = ± 4, y = ± 2

∴ the required points are ( x, y ) ≡ ( 4, 2 ), ( - 4, -2 ) .............. Ans.

...................................................................................................................

Happy To Help !

.....................................................................................................................

- mauriceLv 69 years ago
implicit differentiation ok

plug in y = 0 wrong

dy/dx and look for values of x that make the ans infinity