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Question곡선 y=4xy=\frac{4}{x} 위의 점 A(1,4)와 B(t, 4t\frac{4}{t})를 지나는 직선의 삼각형 OPB 넓이 S(t)S(t)의 lim⁡t→∞S(t)\lim _{t \rightarrow \infty} S(t) 값은?

Studdy Solution

STEP 1

Assumptions1. The curve is given by y=4xy=\frac{4}{x} . The points A and B on the curve are given by A(1,4)\mathrm{A}(1,4) and B(t,4t)\mathrm{B}\left(t, \frac{4}{t}\right) respectively, where t>1t>1
3. The line passing through points A and B intersects the x-axis at point4. The area of triangle OPB is denoted by (t)(t)5. The point O is the origin (0,0)

STEP 2

First, we need to find the equation of the line AB. The slope of the line AB can be calculated using the formulalope=y2−y1x2−x1lope = \frac{y2 - y1}{x2 - x1}

STEP 3

Now, plug in the given values for the coordinates of points A and B to calculate the slope.
lope=/t−t−1lope = \frac{/t -}{t -1}

STEP 4

implify the slope expression.
lope=4(1−t)t(t−1)=4tlope = \frac{4(1 - t)}{t(t -1)} = \frac{4}{t}

STEP 5

The equation of the line AB can be written in the form y−y1=m(x−x1)y - y1 = m(x - x1) where m is the slope and (x1,y1)(x1, y1) is a point on the line. Using point A(1,4) and the calculated slope, the equation of the line isy−4=4t(x−1)y -4 = \frac{4}{t}(x -1)

STEP 6

implify the equation of the line to find the x-intercept (point).
y=4tx+4−4t=4tx+4(1−1t)y = \frac{4}{t}x +4 - \frac{4}{t} = \frac{4}{t}x +4(1 - \frac{1}{t})

STEP 7

Setting y=0y=0 to find the x-coordinate of point.
0=4tx+4(1−1t)0 = \frac{4}{t}x +4(1 - \frac{1}{t})

STEP 8

olving the above equation for x gives the x-coordinate of point.
x=−t(1−1t)=t−1x = -t(1 - \frac{1}{t}) = t -1

STEP 9

Now, we have the coordinates of points O, B and. We can calculate the area of triangle OPB using the formula(t)=2×OP×BP(t) = \frac{}{2} \times OP \times BP

STEP 10

Substitute the lengths of OP and BP into the formula. OP is the x-coordinate of point, and BP is the y-coordinate of point B.
(t)=2×(t−)×4t(t) = \frac{}{2} \times (t -) \times \frac{4}{t}

STEP 11

implify the expression for (t)(t).
(t)=(−t)(t) =( - \frac{}{t})

STEP 12

Now, we need to find the limit of (t)(t) as tt approaches infinity.
lim⁡t→∞(t)=lim⁡t→∞2(−t)\lim{t \rightarrow \infty}(t) = \lim{t \rightarrow \infty}2( - \frac{}{t})

STEP 13

Calculate the limit.
lim⁡t→∞(t)=2(−0)=2\lim{t \rightarrow \infty}(t) =2( -0) =2The limit of (t)(t) as tt approaches infinity is2.

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