Answer

Verified

86.7k+ views

**Hint:**If \[a\], \[b\] and \[c\] are the consecutive terms of an AP, then using the properties of arithmetic progression, \[b-a = c-b\].

Also, the area of the triangle with the vertices \[(x_{1},y_{1})\], \[(x_{2},y_{2})\] and \[(x_{3},y_{3})\] is given by the formula \[A = \dfrac{1}{2}[x_{1}(y_{2}-y_{3}) + x_{2}(y_{3}-y_{1}) + x_{3}(y_{1}-y_{2})]\].

**Complete step-by-step answer:**

Given the terms \[3k-2, 2k^{2}-5k+8\] and \[4k+3\] are the consecutive terms of an AP.

Using the properties of arithmetic progression, the difference between the consecutive terms are equal.

This implies,

\[2k^{2}-5k+8-(3k-2) = 4k+3-(2k^{2}-5k+8)\]

Solving them as follows:

\[\begin{align*}2k^{2}-5k+8-(3k-2) &= 4k+3-(2k^{2}-5k+8)\\ 2k^{2}-8k+10 &= -2k^{2}+9k-5\\

4k^{2}-17k+15 &= 0\\ 4k^{2}-12k-5k+15 &=0\\ 4k(k-3)+3(k-3) &=0\\ (k-3)(4k+3) &=0\\ k &=3, -\dfrac{3}{4}\end{align*}\]

So, the value of \[k = 3, -\dfrac{3}{4}\].

Now, the vertices of the triangle are \[(k,2)\], \[(3k,2)\] and \[(2,5)\], and its area is \[6\,\text{sq.u}\].

So, substituting the values into the formula for the area of the triangle, \[A = \dfrac{1}{2}[x_{1}(y_{2}-y_{3}) + x_{2}(y_{3}-y_{1}) + x_{3}(y_{1}-y_{2})]\], it gives,

\[\begin{align*}6 &= \dfrac{1}{2}[k(2-5)+3k(5-2)+2(2-2)]\\ 12 &= -3k+9k\\ 12 &= 6k\\ k&= 2\end{align*}\]

Therefore, the value of \[k\] is 2.

**Note:**The area of the triangle can also be evaluated using the lengths of the sides of the triangle, and then using Heron's formula to calculate the area.

Recently Updated Pages

Name the scale on which the destructive energy of an class 11 physics JEE_Main

Write an article on the need and importance of sports class 10 english JEE_Main

Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main

Choose the one which best expresses the meaning of class 9 english JEE_Main

What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main

A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main

Other Pages

The thickness of the depletion layer is approximately class 11 physics JEE_Main

Velocity of car at t 0 is u moves with a constant acceleration class 11 physics JEE_Main

If a wire of resistance R is stretched to double of class 12 physics JEE_Main

Derive an expression for maximum speed of a car on class 11 physics JEE_Main

Electric field due to uniformly charged sphere class 12 physics JEE_Main