For what values of p are the points \[\left( {2,1} \right),\left( {p, - 1} \right){\text{ }}and{\text{ }}\left( {1, - 3} \right)\] collinear.
Last updated date: 21st Mar 2023
•
Total views: 304.5k
•
Views today: 8.83k
Answer
304.5k+ views
Hint- In order to solve such types of questions, we will use the basic property that the given points are collinear if and only if the area of the triangle formed by given points should be zero.
Complete step-by-step answer:
Given points are $(2,1),(p, - 1){\text{ and( - 1,3)}}$
Let $A(2,1),B(p, - 1){\text{ andC( - 1,3)}}$ be the coordinates of the straight line.
We know that; the given points are said to be collinear if area of
$\Delta ABC = 0$
So, we will proceed further by determining the area of triangle
$\therefore \Delta = \left| {\begin{array}{*{20}{c}}
2&1&1 \\
p&{ - 1}&1 \\
{ - 1}&3&1
\end{array}} \right|$
Finding the determinant of above equation with the help of 3rd column
$
\Delta = \dfrac{1}{2}\left[ {1\left| {\begin{array}{*{20}{c}}
p&{ - 1} \\
{ - 1}&3
\end{array}} \right| - 1\left| {\begin{array}{*{20}{c}}
2&1 \\
{ - 1}&3
\end{array}} \right| + 1\left| {\begin{array}{*{20}{c}}
2&1 \\
p&{ - 1}
\end{array}} \right|} \right] \\
= \dfrac{1}{2}\left[ {(3p - 1) - 1(6 + 1) + 1( - 2 - p)} \right] \\
= \dfrac{1}{2}\left[ { - 3p - 1 - 7 - 2 - p} \right] \\
= \dfrac{1}{2}[2p - 10] \\
= p - 5 \\
$
Since, area of the triangle should be zero
$
\Rightarrow p - 5 = 0 \\
\Rightarrow p = 5 \\
$
Hence, the value of point p is 5.
Note- To solve these types of questions remember the basic properties of the triangle. In this question we use the basic property of straight line to solve the question i.e. the area of the straight line is zero. We find the area using a matrix formula to determine the area of the triangle.
Complete step-by-step answer:
Given points are $(2,1),(p, - 1){\text{ and( - 1,3)}}$
Let $A(2,1),B(p, - 1){\text{ andC( - 1,3)}}$ be the coordinates of the straight line.
We know that; the given points are said to be collinear if area of
$\Delta ABC = 0$
So, we will proceed further by determining the area of triangle
$\therefore \Delta = \left| {\begin{array}{*{20}{c}}
2&1&1 \\
p&{ - 1}&1 \\
{ - 1}&3&1
\end{array}} \right|$
Finding the determinant of above equation with the help of 3rd column
$
\Delta = \dfrac{1}{2}\left[ {1\left| {\begin{array}{*{20}{c}}
p&{ - 1} \\
{ - 1}&3
\end{array}} \right| - 1\left| {\begin{array}{*{20}{c}}
2&1 \\
{ - 1}&3
\end{array}} \right| + 1\left| {\begin{array}{*{20}{c}}
2&1 \\
p&{ - 1}
\end{array}} \right|} \right] \\
= \dfrac{1}{2}\left[ {(3p - 1) - 1(6 + 1) + 1( - 2 - p)} \right] \\
= \dfrac{1}{2}\left[ { - 3p - 1 - 7 - 2 - p} \right] \\
= \dfrac{1}{2}[2p - 10] \\
= p - 5 \\
$
Since, area of the triangle should be zero
$
\Rightarrow p - 5 = 0 \\
\Rightarrow p = 5 \\
$
Hence, the value of point p is 5.
Note- To solve these types of questions remember the basic properties of the triangle. In this question we use the basic property of straight line to solve the question i.e. the area of the straight line is zero. We find the area using a matrix formula to determine the area of the triangle.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
