1. Another Differentiation

Find the gradient of the curve (1+x) (2+y) = x^2 + y^2 at each of the two points where the curve meets the y-axis.

Show also that there are two points at which the tangents to this curve are parallel to the y-axis.

I’ve done 1st part of this question. But how can I solve 2nd part?

2. Originally Posted by geton
Find the gradient of the curve (1+x) (2+y) = x^2 + y^2 at each of the two points where the curve meets the y-axis.

Show also that there are two points at which the tangents to this curve are parallel to the y-axis.

I’ve done 1st part of this question. But how can I solve 2nd part?
Find the derivative:
$(1 + x)(2 + y) = x^2 + y^2$

$(1)(2 + y) + (1 + x) \frac{dy}{dx} = 2x + 2y \frac{dy}{dx}$

$2 + y - 2x = [-(1 + x) + 2y] \frac{dy}{dx}$

$\frac{dy}{dx} = \frac{-2x + y + 2}{-x + 2y - 1}$

Now, you are looking for tangents where the slope is undefined. The derivative is undefined for
$-x + 2y - 1 = 0$

$y = \frac{1}{2}x + \frac{1}{2}$

So any points where the tangent line is parallel to the y-axis is going to be on this line. Thus find the intersections of this line with the given curve.

-Dan

3. Thank you so much 'topsquark' for help.

4. Originally Posted by topsquark
[snip]

$\frac{dy}{dx} = \frac{-2x + y + 2}{-x + 2y - 1}$

Now, you are looking for tangents where the slope is undefined. The derivative is undefined for
$-x + 2y - 1 = 0$

$y = \frac{1}{2}x + \frac{1}{2}$

So any points where the tangent line is parallel to the y-axis is going to be on this line. Thus find the intersections of this line with the given curve.

-Dan
I'd just like to add a little something here:

It's trivially obvious in this instance that when the denominator is zero, the numerator is NON-zero.

However, there can be times when some of the solutions that make the denominator equal to zero also make the numerator equal to zero. At such points the derivative becomes indeterminant: $\frac{dy}{dx} \rightarrow \frac{0}{0}$. The tangent at these points is NOT necessarily vertical (and there may even be more than one tangent). Such singular points are called multiple points - there are three types:

1. Cusp.

2. Crunode.

3. Acnode.

This existence of a multiple point should always be considered.

A simple example: $y^2 = x^3 + ax^2$.

a = 0: The origin is a cusp.

a > 0: The origin is a crunode.

a < 0: The origin is an acnode.

By the way, a crunode is a point where the curve intersects itself.