Question

Consider the following statements :
$S_1$ : In every triangle with usual notations $R\ge 2r$
$S_2$ : For every hyperbola, there exists alteast one point, from where mutually perpendicular tangents can be drawn to it
$S_3$ : Mutually exclusive events with non-zero probabilities can never be independent.
$S_4$ : Product of lengths of perpendiculars drawn from foci upon any tangent of an ellipse is equal to square of its semi-major axis.
State, in order, whether $S_1$, $S_2$, $S_3$ or $S_4$ are true or false

• A. FFTF
• B. FFTT
• C. TTTF
• D. TFTF

Answer 1

The correct option is D TFTF

$S_1:r=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

$\therefore \frac{r}{R}\le 4\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}$

(When $A=B=C={60}^o$) $\Rightarrow R\ge 2r$
$S_2:\frac{x^2}{9}-\frac{y^2}{16}=1$
No such point is possible when $a<b$
$S_3:P(A\cap B)\ne P\left(A\right).P\left(B\right)$
$\therefore P(A\cap B)=0$ but $P\left(A\right).P\left(B\right)\ne 0$
$S_4:$ it is equal to square of semi-minor axis.