Answer

Verified

429.3k+ views

Hint: Check collinearity of the given 3 points by using section formula.

The given points can be rewritten in simpler terms as

$P=\left( -\sin \left( \beta -\alpha \right),-\cos \beta \right)=\left( {{x}_{1}},{{y}_{1}} \right)\cdots \cdots

\cdots \left( i \right)$

$Q=\left( \cos \left( \beta -\alpha \right),\sin \beta \right)=\left( {{x}_{2}},{{y}_{2}} \right)\cdots \cdots

\cdots \left( ii \right)$

Let the coordinates of the third point $R=\left( \cos \left( \beta -\alpha +\theta \right),\sin \left( \beta -

\theta \right) \right)=\left( {{x}_{3}},{{y}_{3}} \right)$. The ${{x}_{3}}$ coordinate can be simplified as,

${{x}_{3}}=\left( \cos \left( \beta -\alpha +\theta \right) \right)=\cos \left[ \left( \beta -\alpha

\right)+\theta \right]$

Applying the expansion $\cos \left( a+b \right)=\cos a\cos b-\sin a\sin b$,

${{x}_{3}}=\cos \left[ \left( \beta -\alpha \right)+\theta \right]=\cos \left( \beta -\alpha \right)\cos

\theta -\sin \left( \beta -\alpha \right)\sin \theta $

Substituting the corresponding terms from equations $\left( i \right)$ and $\left( ii \right)$,

${{x}_{3}}=\cos \left[ \left( \beta -\alpha \right)+\theta \right]={{x}_{2}}\cos \theta +{{x}_{1}}\sin \theta

$

Now, the ${{y}_{3}}$ coordinate can be simplified as,

${{y}_{3}}=\sin \left( \beta -\theta \right)$

Applying the expansion $\sin \left( a-b \right)=\sin a\cos b-\cos a\sin b$,

${{y}_{3}}=\sin \left( \beta -\theta \right)=\sin \beta \cos \theta -\cos \beta \sin \theta $

Substituting the corresponding terms from equations $\left( i \right)$ and $\left( ii \right)$,

${{y}_{3}}=\sin \left( \beta -\theta \right)={{y}_{2}}\cos \theta +{{y}_{1}}\sin \theta $

So, therefore the third point can be written as,

$R=\left( {{x}_{2}}\cos \theta +{{x}_{1}}\sin \theta ,{{y}_{2}}\cos \theta +{{y}_{1}}\sin \theta \right)\cdots

\cdots \cdots \left( iii \right)$

Consider the line with endpoints PQ. Also consider the point R that lies on the line diving it in the ratio as

below,

Using the section formula, the coordinates of point R can be obtained as,

$R=\left( \dfrac{{{x}_{1}}\cos \theta +{{x}_{2}}\sin \theta }{\sin \theta +\cos \theta },\dfrac{{{y}_{1}}\cos

\theta +{{y}_{2}}\sin \theta }{\sin \theta +\cos \theta } \right)$

From equation $\left( iii \right)$, we have the coordinates of R as $\left( {{x}_{2}}\cos \theta

+{{x}_{1}}\sin \theta ,{{y}_{2}}\cos \theta +{{y}_{1}}\sin \theta \right)$. Comparing this with the above

coordinates, it is clear that the form of the coordinates is not the same.

Therefore, the point R will not lie on the line PQ. It means that the points P, Q and R are not collinear.

Hence, we obtain the correct answer as option (d).

Note: The problem can be solved by applying the condition for collinear points. To check if the points P,

Q and R lie on the same line, consider that point Q lies on line PR. Then, the slope of line PQ and slope of

line QR must be equal for the points to be collinear.

The given points can be rewritten in simpler terms as

$P=\left( -\sin \left( \beta -\alpha \right),-\cos \beta \right)=\left( {{x}_{1}},{{y}_{1}} \right)\cdots \cdots

\cdots \left( i \right)$

$Q=\left( \cos \left( \beta -\alpha \right),\sin \beta \right)=\left( {{x}_{2}},{{y}_{2}} \right)\cdots \cdots

\cdots \left( ii \right)$

Let the coordinates of the third point $R=\left( \cos \left( \beta -\alpha +\theta \right),\sin \left( \beta -

\theta \right) \right)=\left( {{x}_{3}},{{y}_{3}} \right)$. The ${{x}_{3}}$ coordinate can be simplified as,

${{x}_{3}}=\left( \cos \left( \beta -\alpha +\theta \right) \right)=\cos \left[ \left( \beta -\alpha

\right)+\theta \right]$

Applying the expansion $\cos \left( a+b \right)=\cos a\cos b-\sin a\sin b$,

${{x}_{3}}=\cos \left[ \left( \beta -\alpha \right)+\theta \right]=\cos \left( \beta -\alpha \right)\cos

\theta -\sin \left( \beta -\alpha \right)\sin \theta $

Substituting the corresponding terms from equations $\left( i \right)$ and $\left( ii \right)$,

${{x}_{3}}=\cos \left[ \left( \beta -\alpha \right)+\theta \right]={{x}_{2}}\cos \theta +{{x}_{1}}\sin \theta

$

Now, the ${{y}_{3}}$ coordinate can be simplified as,

${{y}_{3}}=\sin \left( \beta -\theta \right)$

Applying the expansion $\sin \left( a-b \right)=\sin a\cos b-\cos a\sin b$,

${{y}_{3}}=\sin \left( \beta -\theta \right)=\sin \beta \cos \theta -\cos \beta \sin \theta $

Substituting the corresponding terms from equations $\left( i \right)$ and $\left( ii \right)$,

${{y}_{3}}=\sin \left( \beta -\theta \right)={{y}_{2}}\cos \theta +{{y}_{1}}\sin \theta $

So, therefore the third point can be written as,

$R=\left( {{x}_{2}}\cos \theta +{{x}_{1}}\sin \theta ,{{y}_{2}}\cos \theta +{{y}_{1}}\sin \theta \right)\cdots

\cdots \cdots \left( iii \right)$

Consider the line with endpoints PQ. Also consider the point R that lies on the line diving it in the ratio as

below,

Using the section formula, the coordinates of point R can be obtained as,

$R=\left( \dfrac{{{x}_{1}}\cos \theta +{{x}_{2}}\sin \theta }{\sin \theta +\cos \theta },\dfrac{{{y}_{1}}\cos

\theta +{{y}_{2}}\sin \theta }{\sin \theta +\cos \theta } \right)$

From equation $\left( iii \right)$, we have the coordinates of R as $\left( {{x}_{2}}\cos \theta

+{{x}_{1}}\sin \theta ,{{y}_{2}}\cos \theta +{{y}_{1}}\sin \theta \right)$. Comparing this with the above

coordinates, it is clear that the form of the coordinates is not the same.

Therefore, the point R will not lie on the line PQ. It means that the points P, Q and R are not collinear.

Hence, we obtain the correct answer as option (d).

Note: The problem can be solved by applying the condition for collinear points. To check if the points P,

Q and R lie on the same line, consider that point Q lies on line PR. Then, the slope of line PQ and slope of

line QR must be equal for the points to be collinear.

Recently Updated Pages

Assertion The resistivity of a semiconductor increases class 13 physics CBSE

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

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Two charges are placed at a certain distance apart class 12 physics CBSE

Difference Between Plant Cell and Animal Cell

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

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

The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE

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

What is BLO What is the full form of BLO class 8 social science CBSE