Question:
If \(r_1,r_2\) and \(r_3\) of a triangle \(ABC\) are in Harmonic progression, then \(a,b,c\) will be in:
If \(r_1,r_2\) and \(r_3\) of a triangle \(ABC\) are in Harmonic progression, then \(a,b,c\) will be in:
Show Hint
If three quantities are in H.P., then their reciprocals are in A.P. For exradii, use \(r_1=\frac{\Delta}{s-a}\), \(r_2=\frac{\Delta}{s-b}\), and \(r_3=\frac{\Delta}{s-c}\).
Updated On: Jun 18, 2026
- Arithmetic progression
- Geometric progression
- Harmonic progression
- Arithmetico-Geometric progression
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Use exradius formulas.
For a triangle \(ABC\), \[ r_1=\frac{\Delta}{s-a},\quad r_2=\frac{\Delta}{s-b},\quad r_3=\frac{\Delta}{s-c} \]
Step 2: Use the condition of harmonic progression.
Since \(r_1,r_2,r_3\) are in H.P., their reciprocals are in A.P.
So, \[ \frac{1}{r_1},\frac{1}{r_2},\frac{1}{r_3} \] are in A.P.
Now, \[ \frac{1}{r_1}=\frac{s-a}{\Delta},\quad \frac{1}{r_2}=\frac{s-b}{\Delta},\quad \frac{1}{r_3}=\frac{s-c}{\Delta} \] Therefore, \[ s-a,\quad s-b,\quad s-c \] are in A.P.
Step 3: Apply A.P. condition.
For three terms in A.P., \[ 2(s-b)=(s-a)+(s-c) \] \[ 2s-2b=2s-a-c \] \[ a+c=2b \] This means \(a,b,c\) are in arithmetic progression.
Step 4: Final conclusion.
Therefore, \[ \boxed{\text{Arithmetic progression}} \]
For a triangle \(ABC\), \[ r_1=\frac{\Delta}{s-a},\quad r_2=\frac{\Delta}{s-b},\quad r_3=\frac{\Delta}{s-c} \]
Step 2: Use the condition of harmonic progression.
Since \(r_1,r_2,r_3\) are in H.P., their reciprocals are in A.P.
So, \[ \frac{1}{r_1},\frac{1}{r_2},\frac{1}{r_3} \] are in A.P.
Now, \[ \frac{1}{r_1}=\frac{s-a}{\Delta},\quad \frac{1}{r_2}=\frac{s-b}{\Delta},\quad \frac{1}{r_3}=\frac{s-c}{\Delta} \] Therefore, \[ s-a,\quad s-b,\quad s-c \] are in A.P.
Step 3: Apply A.P. condition.
For three terms in A.P., \[ 2(s-b)=(s-a)+(s-c) \] \[ 2s-2b=2s-a-c \] \[ a+c=2b \] This means \(a,b,c\) are in arithmetic progression.
Step 4: Final conclusion.
Therefore, \[ \boxed{\text{Arithmetic progression}} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET Trigonometry Questions
- If $\sin \theta + \cos \theta = \sqrt{2} \cos \theta$, then $\cos \theta - \sin \theta =$
- The maximum value of $3 \sin x + 4 \cos x + 5$ is:
- If $\tan^{-1}(x) + \tan^{-1}(y) + \tan^{-1}(z) = \frac{\pi}{2}$, then $xy + yz + zx =$
- If $\sinh x = \frac{3}{4}$, then $\cosh 2x =$
- In a triangle $ABC$, if $a = 13$, $b = 14$, $c = 15$, then the area of the triangle is:
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions