Question

Statement$-1$ : If side $BC$ and the ratio $\frac{r_2}{r_3}$ of an acute angled triangle is given, then the locus of point $A$ is a hyperbola.

Statement$-2$: If the base of a triangle is given and difference of two variable sides is constant (less than the base), then locus of variable vertex is a hyperbola .

• A. Both statements are true and statement $-2$ is the correct explanation of statement $-1$.
• B. Both statements are true but statement $-2$ is NOT the correct explanation of statement $-1$.
• C. Statement$-1$ is true and statement $-2$ is false
• D. Statement$-1$ is false and statement $-2$ is true.

Answer 1

The correct option is A Both statements are true and statement $-2$ is the correct explanation of statement $-1$.

Given $a$ (length of side $BC$) is constant and $\frac{r_2}{r_3}=k$
(where $k$ is constant).
Now,
$\frac{r_2}{r_3}=k$
$\Rightarrow \frac{r_2-r_3}{r_2+r_3}=\frac{k-1}{k+1}$
$\Rightarrow b-c=a\left(\frac{k-1}{k+1}\right)=$ constant
Therefore, locus of vertex $A$ is hyperbola.
Clearly, $a\left(\frac{k-1}{k+1}\right)<a$
Hence both statements are true and statement $-2$ is the correct explanation of statement $-1$.