Find the slope of tangent question (1 Viewer)

GiraffeTD · May 22, 2016

Hi everyone,

I was just wondering if anyone could solve this question with working out?

For the function x^2 + y^2 = 81, find the slope of the tangent to the curve at x=-7.4.

Thank you!

InteGrand · May 22, 2016

GiraffeTD said:
Hi everyone,

I was just wondering if anyone could solve this question with working out?

For the function x^2 + y^2 = 81, find the slope of the tangent to the curve at x=-7.4.

Thank you!

$\noindent At the point $x=-7.4$, you can solve to find the values of $y$ from the circle's equation (note it is a cirlce). Then note that at any point $(x,y)$ on the circle (with a defined slope of normal), the normal is just the line through the origin to the point $(x,y)$, which has slope $\frac{y}{x}$ (due to geometric properties of the circle). So the slope of the \emph{tangent} at any point $(x,y)$ on the circle is $-\frac{x}{y}$, for $y\neq 0$ (at $y=0$, the tangents are vertical).$

si2136 · May 22, 2016

InteGrand said:
$\noindent At the point $x=-7.4$, you can solve to find the values of $y$ from the circle's equation (note it is a cirlce). Then note that at any point $(x,y)$ on the circle (with a defined slope of normal), the normal is just the line through the origin to the point $(x,y)$, which has slope $\frac{y}{x}$ (due to geometric properties of the circle). So the slope of the \emph{tangent} at any point $(x,y)$ on the circle is $-\frac{x}{y}$, for $y\neq 0$ (at $y=0$, the tangents are vertical).$

This is correct, however OP must know that the equation of the slope is y/x due to Implicit Differentiation.

Any point P: (x,y) on the circle satisfies the relation

x^2 + y^2 = 1

Hence, we get that

2x+2y dy/dx = 0 ⟹ dy/dx = −x/y

leehuan · May 22, 2016

Technically because this is a 2U question you would go abouts the old fashioned way with the chain rule.

Just gotta pick the correct case; positive or negative root.

But since OP did not specify a y-coordinate we lack information. At the point x=-7.4 there's two tangents because two y coordinates correspond go x=-7.4

dan964 · May 22, 2016

leehuan said:
Technically because this is a 2U question you would go abouts the old fashioned way with the chain rule.

Just gotta pick the correct case; positive or negative root.

But since OP did not specify a y-coordinate we lack information. At the point x=-7.4 there's two tangents because two y coordinates correspond go x=-7.4

Maybe there are just two solutions then

ThreeSciences · May 28, 2016

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Find the slope of tangent question (1 Viewer)

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