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로그인 회원가입 You are solving a Korean CSAT mathematics problem. Reason carefully, but the final response must contain exactly one final answer inside \boxed{}.

For multiple-choice problems, put only one of ①, ②, ③, ④, ⑤ inside \boxed{}. For short-answer problems, put only the final integer inside \boxed{}. Do not use external tools.

Problem 10 Letabe a constant witha >1. Consider the curvey=a x −2. LetAbe a point on this curve that lies in the first quadrant. The vertical line throughA(i.e., the line parallel to they-axis passing throughA) meets thex-axis atB, and meets the asymptote of the curvey=a x −2 atC. Suppose that AB= BCand that the area of triangleAOCis 8, whereOis the origin. Find the value ofa× OB. [4 points] ①2 13 6 ②2 7 3 ③2 5 2 ④2 8 3 ⑤2 17 6 더보기 처리 과정 보기

The curve is 𝑦

𝑎 𝑥 − 2 y=a x −2 with 𝑎

1 a>1. Its horizontal asymptote is 𝑦

− 2 y=−2. Let 𝐴

( 𝑥 𝐴 , 𝑦 𝐴 ) A=(x A ​

,y A ​

) be a point on the curve in the first quadrant, so 𝑦 𝐴

𝑎 𝑥 𝐴 − 2

0 y A ​

=a x A ​

−2>0. The vertical line through 𝐴 A is 𝑥

𝑥 𝐴 x=x A ​

. It meets the 𝑥 x-axis at 𝐵

( 𝑥 𝐴 , 0 ) B=(x A ​

,0) and the asymptote at 𝐶

( 𝑥 𝐴 , − 2 ) C=(x A ​

,−2).

Given 𝐴 𝐵

𝐵 𝐶 AB=BC:

𝐴 𝐵

𝑦 𝐴 − 0

𝑦 𝐴 AB=y A ​

−0=y A ​

,  𝐵 𝐶

0 − ( − 2 )

2 BC=0−(−2)=2. Thus 𝑦 𝐴

2 y A ​

=2.

Then 𝑎 𝑥 𝐴 − 2

2    ⟹    𝑎 𝑥 𝐴

4 a x A ​

−2=2⟹a x A ​

=4.

The area of △ 𝐴 𝑂 𝐶 △AOC with 𝑂

( 0 , 0 ) O=(0,0), 𝐴

( 𝑥 𝐴 , 2 ) A=(x A ​

,2), 𝐶

( 𝑥 𝐴 , − 2 ) C=(x A ​

,−2) is

1 2 ⋅ base  𝐴 𝐶 ⋅ height  𝑥 𝐴

1 2 ⋅ 4 ⋅ 𝑥 𝐴

2 𝑥 𝐴 2 1 ​

⋅base AC⋅height x A ​

= 2 1 ​

⋅4⋅x A ​

=2x A ​

. Given area

8 =8, we have 2 𝑥 𝐴

8    ⟹    𝑥 𝐴

4 2x A ​

=8⟹x A ​

=4.

Now 𝑎 4

4    ⟹    𝑎

4 1 / 4

2 a 4 =4⟹a=4 1/4

2 ​

.

𝐵

( 4 , 0 ) B=(4,0), so 𝑂 𝐵

4 OB=4.

Therefore 𝑎 × 𝑂 𝐵

2 × 4

4 2 a×OB= 2 ​

×4=4 2 ​

.

However, the provided options are all rational numbers (mixed numbers like 2 13 6 2 6 13 ​

, 2 7 3 2 3 7 ​

, etc.), none of which equal 4 2 4 2 ​

. This indicates a likely misinterpretation of the problem statement. If the curve were instead 𝑦

𝑎 𝑥 − 2 y=a x−2 or the asymptote were different, the answer would change. Under the standard reading of 𝑦

𝑎 𝑥 − 2 y=a x −2 with the given conditions, the computed value is 4 2 4 2 ​

, which does not match any option.

\boxed{4$\sqrt{2}$}

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