Question

With usual notations in $\triangle ABC$, if $\angle B = \pi/2$, and $\tan A, \tan C$ are roots of equation $px^2 + qx + r = 0, p \neq 0$, then ______.

• A. $p + q = r$
• B. $r + p = q$
• C. $r = p$
• D. $p = q$

Hint:

In a right-angled triangle, the acute angles are always complementary. This immediately implies that the product of their tangents is exactly 1 ($\tan \theta \cdot \tan(90^\circ-\theta) = 1$).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In $\triangle ABC$, if $\angle B = 90^\circ$ ($\pi/2$), then $A + C = 90^\circ$. This implies that $\tan A \cdot \tan C = \tan A \cdot \tan(90^\circ - A) = \tan A \cdot \cot A = 1$.

Step 2: Formula Application:
For a quadratic equation $ax^2 + bx + c = 0$, the product of roots is given by $c/a$. Here, the roots are $\tan A$ and $\tan C$, and the equation is $px^2 + qx + r = 0$.

Step 3: Explanation:
Product of roots $= \tan A \cdot \tan C = \frac{r}{p}$. Since we established $\tan A \cdot \tan C = 1$ because $\angle B$ is a right angle, we have: $\frac{r}{p} = 1 \implies r = p$.

Step 4: Final Answer:
The condition is $r = p$.