Answer

Verified

438.9k+ views

**Hint:**This problem comes under co–ordinate geometry. We need to use section formulas internally to solve this problem. The points are collinear which means all points lie on the same line. Since the points are collinear initially we will solve \[l:1\] ratios and then we get the ratio value of \[l\] with that we can find $\lambda $ and then basic mathematical calculation and complete step by step explanation.

**Formula used:**Section formula internally in the ratio \[l:m\]

\[(x,y) = \left( {\dfrac{{l{x_2} + m{x_1}}}{{l + m}},\dfrac{{ly{}_2 + m{y_2}}}{{l + m}}} \right)\]

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

Consider the given collinear points as $A(6, - 1,2)$, $B(8, - 7,\lambda )$, $C(5,2,4)$

Let us assume

$A(6, - 1,2) = \left( {{x_1},{y_1},{z_1}} \right)$

$B(8, - 7,\lambda ) = \left( {{x_2},{y_2},{z_2}} \right)$

$C(5,2,4) = \left( {{x_3},{y_3},{z_3}} \right)$

Use the formula mentioned in formula used and take only $x$ – axis coordinates only

\[ \Rightarrow x = \dfrac{{l{x_2} + m{x_1}}}{{l + m}}\]

Set \[B\] cuts \[AC\] in the ratio \[l:1\]and then substitute the value of $x$ on their respective places

\[ \Rightarrow {x_2} = \dfrac{{l{x_3} + m{x_1}}}{{l + m}}\]

Substituting the values ${x_3} = 5$ and ${x_1} = 6$,

\[ \Rightarrow 8 = \dfrac{{l(5) + 1(6)}}{{l + 1}}\]

Simplifying we get,

\[ \Rightarrow 8 = \dfrac{{5l + 6}}{{l + 1}}\]

Now re-arrange the values, we get

\[ \Rightarrow 8(l + 1) = 5l + 6\]

Multiplying the terms,

\[ \Rightarrow 8l + 8 = 5l + 6\]

Now separate \[l\],

\[8l - 5l = 6 - 8\]

Solving for \[l\] we get,

\[ \Rightarrow 3l = - 2\]

\[l = \dfrac{{ - 2}}{3}\]

Hence, \[B\] cuts \[AC\] in the ratio \[\dfrac{{ - 2}}{3}:1\]

Similarly, Use the formula mentioned in formula used and take only \[z\] – axis co-ordinates only because we need to find \[\lambda \]

\[ \Rightarrow {z_2} = \dfrac{{l{z_3} + m{z_1}}}{{l + m}}\]

Substituting the values,

\[ \Rightarrow \lambda = \dfrac{{\left( {\dfrac{{ - 2}}{3}} \right)(4) + (2)}}{{\left( {\dfrac{{ - 2}}{3}} \right) + 1}}\]

Simplifying we get,

\[ \Rightarrow \lambda = \dfrac{{\dfrac{{ - 8}}{3} + 2}}{{\dfrac{{ - 2}}{3} + 1}}\]

Now, we are going to do fraction addition, we get

\[ \Rightarrow \lambda = \dfrac{{\dfrac{{ - 8}}{3} + \dfrac{6}{3}}}{{\dfrac{{ - 2}}{3} + \dfrac{3}{3}}}\]

The denominators are same, so simplifying we get,

\[ \Rightarrow \lambda = \dfrac{{\dfrac{{ - 8 + 6}}{3}}}{{\dfrac{{ - 2 + 3}}{3}}}\]

Adding the terms we get,

\[ \Rightarrow \lambda = \dfrac{{\dfrac{{ - 2}}{3}}}{{\dfrac{1}{3}}}\]

Taking reciprocal to solve,

\[ \Rightarrow \lambda = \dfrac{{ - 2}}{3} \times \dfrac{3}{1}\]

By cancelling \[3\], we get

\[\lambda = - 2\]

**$\therefore $ Thus the value of \[\lambda \] is \[ - 2\].**

**Note:**This kind of problem needs attention on the section formula. We need to know about the collinear points so that we can find the ratio. For that we are able to solve the equation in similar methods. First we find the ratio of \[l\] and then with that again using the section formula we find \[\lambda \]. Like this kind of problem to find a missing point we have to use this kind of method which is easy to solve and makes the solution simple.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

How do you graph the function fx 4x class 9 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What organs are located on the left side of your body class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell