2 years ago

True or False To find the circumference of a circle you multiply the diameter by pi

1 answer:

4 0

Answer:

false

Step-by-step explanation:

Formula: c=2 $\pi$ r

Circumference= two times pi times radius

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Differentiate both sides of the equation of the circle with respect to $x$ , treating $y=y(x)$ as a function of $x$ :

$x^2+y^2=5\implies2x+2y\dfrac{\mathrm dy}{\mathrm dx}=0\implies\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac xy$

This gives the slope of any line tangent to the circle at the point $(x,y)$ .

Rewriting the given line in slope-intercept form tells us its slope is

$x=2y+5\implies y=\dfrac12x-\dfrac52\implies\mathrm{slope}=\dfrac12$

In order for this line to be tangent to the circle, it must intersect the circle at the point $(x,y)$ such that

$-\dfrac xy=\dfrac12\implies y=-2x$

In the equation of the circle, we have

$x^2+(-2x)^2=5x^2=5\implies x=\pm1\implies y=\mp2$

If $x=-1$ , then $-1=2y+5\implies y=-3\neq2$ , so we omit this case.

If $x=1$ , then $1=2y+5\implies y=-2$ , as expected. Therefore $x=2y+5$ is a tangent line to the circle $x^2+y^2=5$ at the point (1, -2).

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