In any triangle ABC,r_1 r_2 + r_2 r_3 + r_3 r | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) We know that the exradii are given by $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$.Substituting these into

Vedclass · Accepted Answer

$\frac{\Delta^2}{r^2}$

Vedclass · Answer

(A) We know that the exradii are given by $r_1 = \frac{\Delta}{s-a}$,$r_2 = \frac{\Delta}{s-b}$,and $r_3 = \frac{\Delta}{s-c}$.Substituting these into the expression $r_1 r_2 + r_2 r_3 + r_3 r_1$:$= \frac{\Delta^2}{(s-a)(s-b)} + \frac{\Delta^2}{(s-b)(s-c)} + \frac{\Delta^2}{(s-c)(s-a)}$$= \frac{\Delta^2(s-c + s-a + s-b)}{(s-a)(s-b)(s-c)}$Since $2s = a+b+c$,we have $s-a+s-b+s-c = 3s - (a+b+c) = 3s - 2s = s$.Also,we know that $\Delta^2 = s(s-a)(s-b)(s-c)$,so $(s-a)(s-b)(s-c) = \frac{\Delta^2}{s}$.Substituting these values:$= \frac{\Delta^2 \cdot s}{\frac{\Delta^2}{s}} = s^2$Since $r = \frac{\Delta}{s}$,we have $s = \frac{\Delta}{r}$,so $s^2 = \frac{\Delta^2}{r^2}$.

In any $\triangle ABC$,$r_1 r_2 + r_2 r_3 + r_3 r_1$ is equal to

Similar Questions

$A$ bullet of mass $m$ strikes a block of mass $M$ connected to a light spring of stiffness $k,$ with a speed $v_0.$ If the bullet gets embedded in the block,then the maximum compression in the spring is

The expression $[x + (x^3 - 1)^{1/2}]^5 + [x - (x^3 - 1)^{1/2}]^5$ is a polynomial of degree :

The centre of mass of two masses $m$ and $m'$ moves by a distance $\frac{x}{5}$ when mass $m$ is moved by a distance $x$ and $m'$ is kept fixed. The ratio $\frac{m'}{m}$ is

If $f(x) = \begin{cases} x \sin \frac{1}{x}, & x \ne 0 \\ k, & x = 0 \end{cases}$ is continuous at $x = 0$,then the value of $k$ is

The set of values of $x$ for which the inequalities $x^2-3x-10 < 0$ and $10x-x^2-16 > 0$ hold simultaneously,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module