JEE Advanced · Mathematics · 9. Straight Lines
Consider three points \(P=(-\sin (\beta-\alpha),-\cos \beta), Q=(\cos (\beta-\alpha), \sin \beta)\) and \(R=(\cos (\beta-\alpha+\theta), \sin (\beta-\theta)\), where \(0 < \alpha, \beta, \theta < \frac{\pi}{4}\). Then,
- A
\(P\) lies on the line segment \(R Q\)
- B
\(Q\) lies on the line segment \(P R\)
- C
\(R\) lies on the line segment \(Q P\)
- D
\(P, Q, R\) are non- collinear
Answer & Solution
Correct Answer
(D)
\(P, Q, R\) are non- collinear
Step-by-step Solution
Detailed explanation
For collinear points
\[
\Delta=\left|\begin{array}{ccc}
-\sin (\beta-\alpha) & -\cos \beta & 1 \\
\cos (\beta-\alpha) & \sin \beta & 1 \\
\cos (\beta-\alpha+\theta) & \sin (\beta-\theta) & 1
\end{array}\right|
\]
Clearly, \(\Delta \neq 0\) for any value of \(\alpha, \beta, \theta\), hence points are non-collinear.
\[
\Delta=\left|\begin{array}{ccc}
-\sin (\beta-\alpha) & -\cos \beta & 1 \\
\cos (\beta-\alpha) & \sin \beta & 1 \\
\cos (\beta-\alpha+\theta) & \sin (\beta-\theta) & 1
\end{array}\right|
\]
Clearly, \(\Delta \neq 0\) for any value of \(\alpha, \beta, \theta\), hence points are non-collinear.
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Mathematics
- Let \(\mathbb{R}\) denote the set of all real numbers. Then the area of the region \(\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x>0, y>\frac{1}{x}, 5 x-4 y-1>0,4 x+4 y-17 <0\right\}\) isJEE Advanced 2025 Hard
- Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\left\{\begin{array}{cc}x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0 \\ 0, & \text { if } x=0\end{array}\right.\) Then which of the following statements is TRUE?JEE Advanced 2024 Hard
- Paragraph:
Let \(p\) be an odd prime number and \(T_p\) be the following set of \(2 \times 2\) matrices
\[
T_p=\left\{A=\left[\begin{array}{ll}
a & b \\
c & a
\end{array}\right] ; a, b, c \in\{0,1,2, \ldots, p-1\}\right\}
\]Question:
The number of \(A\) in \(T_p\) such that \(A\) is either symmetric or skew-symmetric or both, and det \((A)\) is divisible by \(p\) isJEE Advanced 2010 Hard - Let \(a, b, c, p, q\) be real numbers.
Suppose, \(\alpha, \beta\) are the roots of the equation \(x^2+2 p x+q=0\) and \(\alpha, \frac{1}{\beta}\) are the roots of the equation \(a x^2+2 b x+c=0\), where \(\beta^2 \notin\{-1,0,1\}\).
Statement \(1\left(p^2-q\right)\left(b^2-a c\right) \geq 0\).
Statement \(2 b \neq p a\) or \(c \neq q a\).JEE Advanced 2008 Hard - A man walks a distance of 3 units from the origin towards the North-East \(\left(\mathrm{N} 45^{\circ} \mathrm{E}\right)\) direction. From there, he walks a distance of 4 units towards the North-West ( \(\mathrm{N} 45^{\circ} \mathrm{W}\) ) direction to reach a point \(P\). Then, the position of \(P\) in the Argand plane isJEE Advanced 2007 Easy
- Let be a sequence of positive integers in arithmetic progression with common difference . Also, let be a sequence of positive integers in geometric progression with common ratio . If , then the number of all possible values of , for which the equality holds for some positive integer , is _______JEE Advanced 2020 Medium
More PYQs from JEE Advanced
- Two point charges \(-Q\) and \(+Q / \sqrt{3}\) are placed in the \(x y\)-plane at the origin \((0,0)\) and a point \((2,0)\), respectively, as shown in the figure. This results in an equipotential circle of radius \(R\) and potential \(V=0\) in the \(x y\)-plane with its center at \((b, 0)\). All lengths are measured in meters.
The value of is ________ meter.JEE Advanced 2021 Medium - In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle with the horizontal floor. The coefficient of friction between the wall and the ladder is and that between the floor and the ladder is . The normal reaction of the wall on the ladder is and that of the floor is If the ladder is about to slip, then
JEE Advanced 2014 Medium - In the circuit shown, They are connected in series with an a.c. source as shown. Which of the following options is/are correct?
JEE Advanced 2017 Hard - At room temperature, disproportionation of an aqueous solution of in situ generated nitrous acid \(\left(\mathrm{HNO}_2\right)\) gives the speciesJEE Advanced 2024 Easy
- A particle of mass is initially at rest at the origin. It is subjected to a force and starts moving along the x - axis. Its kinetic energy changes with time as where is a positive constant of appropriate dimensions. Which of the following statements is (are) true?JEE Advanced 2018 Medium
- Let be a complex number with non-zero imaginary part. If is a real number, then the value of is ______.JEE Advanced 2022 Hard