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grigory [225]
3 years ago
12

Verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given poi

nt.
I. x2 + xy - y2 = 1, (2,3) (10)
II x2 + y2 = 25, (3,-4)
III. x3y2 = 9, (-1,3)
IV. y2 – 2x – 4y - 1 = 0, (-2, 1)
Mathematics
1 answer:
lara [203]3 years ago
3 0

For each curve, plug in the given point (x,y) and check if the equality holds. For example:

(I) (2, 3) does lie on x^2+xy-y^2=1 since 2^2 + 2*3 - 3^2 = 4 + 6 - 9 = 1.

For part (a), compute the derivative \frac{\mathrm dy}{\mathrm dx}, and evaluate it for the given point (x,y). This is the slope of the tangent line at the point. For example:

(I) The derivative is

x^2+xy-y^2=1\overset{\frac{\mathrm d}{\mathrm dx}}{\implies}2x+x\dfrac{\mathrm dy}{\mathrm dx}+y-2y\dfrac{\mathrm dy}{\mathrm dx}=0\implies\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2x+y}{2y-x}

so the slope of the tangent at (2, 3) is

\dfrac{\mathrm dy}{\mathrm dx}(2,3)=\dfrac74

and its equation is then

y-3=\dfrac74(x-2)\implies y=\dfrac74x-\dfrac12

For part (b), recall that normal lines are perpendicular to tangent lines, so their slopes are negative reciprocals of the slopes of the tangents, -\frac1{\frac{\mathrm dy}{\mathrm dx}}. For example:

(I) The tangent has slope 7/4, so the normal has slope -4/7. Then the normal line has equation

y-3=-\dfrac47(x-2)\implies y=-\dfrac47x+\dfrac{29}7

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shepuryov [24]

Answer:

The number of miles Mark runs in each track be 16 \frac{43}{52}\ miles

Step-by-step explanation:

Let us assume that the number of miles Mark run in each track meet be x.

As given

Mark ran 875 miles this year in the track club.

Mark ran in 52 track meets and ran the same number of miles in each.

Than the equation becomes

52 × x = 875

x = \frac{875}{52}

x = 16 \frac{43}{52}\ miles

Therefore the number of miles Mark runs in each track be 16 \frac{43}{52}\ miles

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The burges are moving across the country .Mr burges leaves 3 hours before mrs.burges. If he averages 50 mph and she averages 70
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10.5 hours

let t be time then distance travelled by both is

Mr Burges ⇔ d = 50t ( distance = speed × time )

Mrs Burges ⇒ d = 70(t - 3)

At the point of overtaking both will have travelled the same distance

equating their distance and solving for t

70(t - 3) = 50t

70t - 210 = 50t ( subtract 50t from both sides )

20t - 210 = 0 ( add 210 to both sides )

20t = 210 ( divide both sides by 20 )

t = 10.5 hours


4 0
3 years ago
A rectangle is 30 cm long and 10 cm wide. What's the ratio of the rectangle's length to its width?
Veseljchak [2.6K]

Answer:

3: 1

Step-by-step explanation:

hope this helps

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Answer:

0.083333333333 or 8.3%

Step-by-step explanation:

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What is 965,000,000,000,000 in scientific notation?
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<h3><u>QUES</u><u>TION</u><u>:</u></h3>

What is 965,000,000,000,000 in scientific notation?

9.65×10−14

9.65×10−12

9.65×1012

9.65×1014

<h3><u>ANSWER</u><u>:</u></h3>

\large \sf \orange{9.65 \times 10 {}^{14} } \sf \: is \: the \: scientific \: notation \: of \: 965000000000000

<u>Addit</u><u>ional</u><u> </u><u>info</u><u>:</u><u> </u>Remember that SCIENTIFIC NOTATION is in the form a × 10^b where 1 ≤ a < 10 and b is an integer.

HOPE THIS HELP YOU! HAVE A NICE DAY

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