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3 years ago

Which of the following points is on a circle if its center is (-13,-12) and a point on the circumference is (-17, -12)?

Mathematics

1 answer:

tia_tia [17]3 years ago

4 0

Answer:

Step-by-step explanation:

Obtain the equation of the circle in standard form

(x - h)² + (y - k)² = r²

where (h, k) are the coordinates of the centre and r is the radius

here (h, k) = (- 13, - 12), thus

(x + 13)² + (y + 12)² = r²

The radius is the distance from the centre (- 13, - 12) to the point on the circumference (- 17, - 12)

Use the distance formula to calculate r

r = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (- 17, - 12) and (x₂, y₂ ) = (- 13, -12)

r = $\sqrt{(-13+17)^2+(-12+12)^2}$ = $\sqrt{16}$ = 4

Hence

(x + 13)² + (y + 12)² = 16 ← in standard form

Substitute the coordinates of each point into the left side of the equation and check

A (- 17, - 13) : (- 4)² + (- 1)² = 16 + 1 = 17 ≠ 16

B (- 9, - 17) : 4² + (- 5)² = 16 + 25 = 41 ≠ 16

C (- 12, 13) : 1² + 25² ≠ 16

D (- 9, - 12) : 4² + 0² = 16

Since (- 9, - 12) satisfies the equation, it is on the circle → D

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2 years ago

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e-lub [12.9K]

Answer:

$(\frac{x^{2}-25}{2})ln(5+x)-\frac{x^{2}}{4}+\frac{5x}{2}+C$

Step-by-step explanation:

Ok, so we start by setting the integral up. The integral we need to solve is:

$\int x ln(5+x)dx$

so according to the instructions of the problem, we need to start by using some substitution. The substitution will be done as follows:

U=5+x

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x=U-5

so when substituting the integral will look like this:

$\int (U-5) ln(U)dU$

now we can go ahead and integrate by parts, remember the integration by parts formula looks like this:

$\int (pq')=pq-\int qp'$

so we must define p, q, p' and q':

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$p'=\frac{1}{U}dU$

$q=\frac{U^{2}}{2}-5U$

q'=U-5

and now we plug these into the formula:

$\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\int \frac{\frac{U^{2}}{2}-5U}{U}dU$

Which simplifies to:

$\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\int (\frac{U}{2}-5)dU$

Which solves to:

$\int (U-5)lnUdU=(\frac{U^{2}}{2}-5U)lnU-\frac{U^{2}}{4}+5U+C$

so we can substitute U back, so we get:

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and now we can simplify:

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