Question

If the feet of the perpendiculars drawn from the point \[ (3,4,5) \] to the \(X\)-, \(Y\)- and \(Z\)-coordinate axes are \(A,B,C\) respectively and the angle between \(AB\) and \(AC\) is \[ \cos^{-1}\left(\frac{9}{a}\right), \] then the value of \(a\) is:

• A. \(5\sqrt{34}\)
• B. \(3\sqrt{34}\)
• C. \(2\sqrt{34}\)
• D. \(\sqrt{34}\)

Hint:

Whenever an angle between two lines or vectors is asked, immediately think of the dot-product formula \[ \cos\theta= \frac{\vec a\cdot\vec b}{|\vec a||\vec b|}. \] It is the fastest method for such problems.

Answer 1

The Correct Option is A

Step 1: Find the feet on the axes.
The foot of the perpendicular from $(3,4,5)$ to the $X$-axis keeps only the $x$-part: $A=(3,0,0)$. Likewise $B=(0,4,0)$ on the $Y$-axis and $C=(0,0,5)$ on the $Z$-axis.

Step 2: Form the two vectors from $A$.
\[ \vec{AB}=B-A=(-3,4,0),\qquad \vec{AC}=C-A=(-3,0,5). \]

Step 3: Take the dot product.
\[ \vec{AB}\cdot\vec{AC}=(-3)(-3)+(4)(0)+(0)(5)=9. \]

Step 4: Find the magnitudes.
\[ |\vec{AB}|=\sqrt{9+16+0}=5,\qquad |\vec{AC}|=\sqrt{9+0+25}=\sqrt{34}. \]

Step 5: Use the cosine formula.
\[ \cos\theta=\frac{\vec{AB}\cdot\vec{AC}}{|\vec{AB}||\vec{AC}|}=\frac{9}{5\sqrt{34}}. \]

Step 6: Compare with the given form.
The angle is written as $\cos^{-1}\!\left(\dfrac{9}{a}\right)$, so $a=5\sqrt{34}$, which is option 1.
\[ \boxed{\,5\sqrt{34}\,} \]