Which of the following correctly shows the length of each radius, the point where the circles intersect, and the equation of the
tangent line at this point?
1 answer:
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To find the equation of a line in slope-intercept form given two points, we must arrange the points this way to find the slope (
):
We can replace the variables in this equation with the values from the points we have been given, (
) and (2, 14). We know that the x-values always come first in an ordered pair, so we can put those in our equation first.
It doesn't really matter which x-value goes in which x-value slot in the equation as long as you match up the y-values in the same fashion. But for the sake of convenience, we will call the 2 in the second ordered pair
and the 
in the first ordered pair 
.
Now, we can match these values in our equation, and while we're at it, we can substitute the appropriate y-values into their places as well.
Now, we can see that we are subtracting negatives in this equation. Remember that whenever you subtract a negative, it is the same as adding a positive. So, this equation could be rewritten as
and we can change the improper fraction in the numerator into a mixed number for ease of addition.
And here, we can add, since it's made simple for us.
Finally, to get the slope we can complete the fraction by dividing.
The slope of this equation,, is 7, and so far, our equation looks like this:
Now, to find y-intercept.
To do this, we simply have to substitute one of the ordered pairs in for the appropriate x- and y-values and solve. To make it easier on ourselves, we can use (2, 14) so we don't have to deal with negative numbers or fractions.
Let us substitute in our values.
Now, we can solve by multiplying the 7 and 2.
Here, we can subtract 14 from both sides, since it is added to both in the equation, and we are left with
which can be flipped around to show
The y-intercept of this line is 0.
Now, we can make this known in our equation like this:
Or, for neatness' sake, we can just say
which is your final equation.
The line through () and (2, 14) in slope-intercept form is <span>.
Hope that helped! =) </span>
We can replace the variables in this equation with the values from the points we have been given, (
It doesn't really matter which x-value goes in which x-value slot in the equation as long as you match up the y-values in the same fashion. But for the sake of convenience, we will call the 2 in the second ordered pair
Now, we can match these values in our equation, and while we're at it, we can substitute the appropriate y-values into their places as well.
Now, we can see that we are subtracting negatives in this equation. Remember that whenever you subtract a negative, it is the same as adding a positive. So, this equation could be rewritten as
and we can change the improper fraction in the numerator into a mixed number for ease of addition.
And here, we can add, since it's made simple for us.
Finally, to get the slope we can complete the fraction by dividing.
The slope of this equation,
Now, to find y-intercept.
To do this, we simply have to substitute one of the ordered pairs in for the appropriate x- and y-values and solve. To make it easier on ourselves, we can use (2, 14) so we don't have to deal with negative numbers or fractions.
Let us substitute in our values.
Now, we can solve by multiplying the 7 and 2.
Here, we can subtract 14 from both sides, since it is added to both in the equation, and we are left with
which can be flipped around to show
The y-intercept of this line is 0.
Now, we can make this known in our equation like this:
Or, for neatness' sake, we can just say
which is your final equation.
The line through (
Hope that helped! =) </span>