Let's solve the problem step by step.
The problem provides information about the angle of elevation of the top of a hill from two points at distances [tex]\( a \)[/tex] and [tex]\( b \)[/tex] from the base of the hill. It states that these angles of elevation are complementary.
1. Understanding Complementary Angles:
- If two angles are complementary, their sum is [tex]\(90^\circ\)[/tex].
- Let's denote these angles as [tex]\(\theta\)[/tex] and [tex]\(90^\circ - \theta\)[/tex].
2. Trigonometric Relationships:
- At distance [tex]\(a\)[/tex], the angle of elevation is [tex]\(\theta\)[/tex].
- At distance [tex]\(b\)[/tex], the angle of elevation is [tex]\(90^\circ - \theta\)[/tex].
3. Tan and Cot Functions:
- The tangent of an angle [tex]\(\theta\)[/tex] is given by [tex]\(\tan(\theta)\)[/tex], which is the ratio of the height (of the hill) to the base (distance [tex]\(a\)[/tex]).
- The cotangent of an angle [tex]\( \theta \)[/tex] is given by [tex]\(\cot(\theta)\)[/tex], which is the reciprocal of the tangent function.
4. Relate tan and cot:
- From the point at distance [tex]\(a\)[/tex], [tex]\(\tan(\theta) = \frac{h}{a}\)[/tex], where [tex]\(h\)[/tex] is the height of the hill.
- From the point at distance [tex]\(b\)[/tex], [tex]\(\tan(90^\circ - \theta) = \cot(\theta) = \frac{b}{h}\)[/tex].
5. Solving the Height of the Hill:
- Since [tex]\(\cot(\theta) = \frac{1}{\tan(\theta)}\)[/tex], we can write: [tex]\(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{b}{h}\)[/tex].
- Rewriting the above equation in terms of height [tex]\(h\)[/tex] and [tex]\(\tan(\theta)\)[/tex], we get [tex]\( \frac{1}{\frac{h}{a}} = \frac{b}{h}\)[/tex].
- This simplifies to [tex]\( \frac{a}{h} = \frac{b}{a} \cdot \tan(\theta)\)[/tex].
6. Expression for Height [tex]\(h\)[/tex]:
- From [tex]\( \tan(\theta) = \frac{h}{a} \)[/tex] and substituting this into [tex]\(\cot(\theta) = \frac{b}{h}\)[/tex], we get:
[tex]\[
\frac{1}{\frac{h}{a}} = \frac{h}{b}
\][/tex]
- Therefore, [tex]\(a = \frac{h^2}{b}\)[/tex].
7. Solving for [tex]\(h\)[/tex]:
- Multiplying both sides by [tex]\(b\)[/tex], we obtain:
[tex]\[
h^2 = a \cdot b
\][/tex]
- Taking the square root of both sides gives:
[tex]\[
h = \sqrt{a \cdot b}
\][/tex]
Hence, the height of the hill is [tex]\(\sqrt{a \cdot b}\)[/tex].
Therefore, the correct answer is [tex]\( \boxed{\sqrt{a b}} \)[/tex].
