A ladder rests against a wall at an angle to the horizontal. Its …

Question

A ladder rests against a wall at an angle $\alpha$ to the horizontal. Its foot is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle $\beta$ with the horizontal. Show that, $\frac{\text{a}}{\text{b}}=\frac{\cos\alpha-\cos\beta}{\sin\beta-\sin\alpha}.$

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Answer

Let $PQ$ be the ladder such that its top $Q$ is on the wall $OQ$ and bottom $P$ is on the ground. The ladder is pulled away from the wall through a distance a, so that its top $Q$ slides and takes position $Q'$. So $PQ = P'Q'$
$\angle\text{OPQ}=\alpha$ and $\angle\text{OP}'\text{Q}'=\beta.$ Let $PQ = h$
We have to prove that,
$\frac{\text{a}}{\text{b}}=\frac{\cos\alpha-\cos\beta}{\sin\beta-\sin\alpha}$
We have the corresponding figure as follows,

We use trigonometric ratios.
In $\triangle\text{POQ}$
$\Rightarrow\ \sin\alpha=\frac{\text{OQ}}{\text{PQ}}$
$\Rightarrow\ \sin\alpha=\frac{\text{b}+\text{y}}{\text{h}}$
And,
$\Rightarrow\ \cos\alpha=\frac{\text{OP}}{\text{PQ}}$
$\Rightarrow\ \cos\alpha=\frac{\text{x}}{\text{h}}$
Again in $\triangle\text{P}'\text{OQ}'$
$\Rightarrow\ \sin\beta=\frac{\text{OQ}'}{\text{P}'\text{Q}'}$
$\Rightarrow\ \sin\beta=\frac{\text{y}}{\text{h}}$
And,
$\Rightarrow\ \cos\beta=\frac{\text{OP}'}{\text{P}'\text{Q}'}$
$\Rightarrow\ \cos\beta=\frac{\text{a}+\text{x}}{\text{h}}$
Now,
$\Rightarrow\ \sin\alpha-\sin\beta=\frac{\text{b}+\text{y}}{\text{h}}-\frac{\text{y}}{\text{h}}$
$\Rightarrow\ \sin\alpha-\sin\beta=\frac{\text{b}}{\text{h}}$
And,
$\Rightarrow\ \cos\beta-\cos\alpha=\frac{\text{a}+\text{x}}{\text{h}}-\frac{\text{x}}{\text{h}}$
$\Rightarrow\ \cos\beta-\cos\alpha=\frac{\text{a}}{\text{h}}$
So,
$\Rightarrow\ \frac{\sin\alpha-\sin\beta}{\cos\beta-\cos\alpha}=\frac{\text{b}}{\text{a}}$
$\Rightarrow\ \frac{\text{a}}{\text{b}}=\frac{\cos\beta-\cos\alpha}{\sin\alpha-\sin\beta}$
$\Rightarrow\ \frac{\text{a}}{\text{b}}=\frac{\cos\alpha-\cos\beta}{\sin\beta-\sin\alpha}$
Hence $\frac{\text{a}}{\text{b}}=\frac{\cos\alpha-\cos\beta}{\sin\beta-\sin\alpha}.$