JEE Mains · Maths · STD 11 - 9. straight line
If \(\mathrm{p}\) and \(\mathrm{q}\) are the lengths of the perpendiculars from the origin on the lines, \(x \operatorname{cosec} \alpha-y \sec \alpha=\operatorname{kcot} 2 \alpha\) and \(x \sin \alpha+y \cos \alpha=k \sin 2 \alpha\) respectively, then \(\mathrm{k}^{2}\) is equal to :
- A \(4 \mathrm{p}^{2}+\mathrm{q}^{2}\)
- B \(2 \mathrm{p}^{2}+\mathrm{q}^{2}\)
- C \(\mathrm{p}^{2}+2 \mathrm{q}^{2}\)
- D \(\mathrm{p}^{2}+4 \mathrm{q}^{2}\)
Answer & Solution
Correct Answer
(A) \(4 \mathrm{p}^{2}+\mathrm{q}^{2}\)
Step-by-step Solution
Detailed explanation
First line is \(\frac{\mathrm{x}}{\sin \alpha}-\frac{\mathrm{y}}{\cos \alpha}=\frac{\mathrm{k} \cos 2 \alpha}{\sin 2 \alpha}\) \(\Rightarrow x \cos \alpha-\operatorname{ysin} \alpha=\frac{\mathrm{k}}{2} \cos 2 \alpha\)…
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 Maths
- If \(\displaystyle\lim_{x \to 2} \dfrac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1} - 1)\log_e(x-1)} = m\), then \(a + b + m\) is equal to :JEE Mains 2026 Hard
- Let \(C\) be the centre of the circle \(x^{2}+y^{2}-x+2 y=\) \(\frac{11}{4}\) and \(P\) be a point on the circle. A line passes through the point \(C\), makes an angle of \(\frac{\pi}{4}\) with the line \(C P\) and intersects the circle at the points \(Q\) and \(R\). Then the area of the triangle \(P Q R\) (in unit \({ }^{2}\) ) is.JEE Mains 2022 Hard
- If the point of intersections of the ellipse \(\frac{ x ^{2}}{16}+\frac{ y ^{2}}{ b ^{2}}=1\) and the circle \(x ^{2}+ y ^{2}=4 b , b > 4\) lie on the curve \(y^{2}=3 x^{2},\) then \(b\) is equal to:JEE Mains 2021 Hard
- The mean and standard deviation of \(50\) observations are \(15\) and \(2\) respectively. It was found that one incorrect observation was taken such that the sum of correct and incorrect observations is \(70\) . If the correct mean is \(16\) , then the correct variance is equal toJEE Mains 2022 Medium
- In a triangle \(ABC,\) right angled at the vertex \(A,\) if the position vectors of \(A, B\) and \(C\) are respectively \(3\hat i\, + \hat j\, - \hat k,\,\, - \hat i\, + 3\hat j\, + p\hat k\) and \(5\hat i\, + q\hat j\, - 4\hat k,\,\) then the point \((p, q)\) lies on a lineJEE Mains 2016 Hard
- If \((a, b, c)\) is the image of the point \((1,2,-3)\) in the line, \(\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1},\) then \(a+b+c\) is equal toJEE Mains 2020 Medium
More PYQs from JEE Mains
- Let \(A\) be a matrix of order \(2 \times 2\), whose entries are from the set \(\{0,1,2,3,4,5\}\). If the sum of all the entries of \(A\) is a prime number \(p , 2< p <8\), then the number of such matrices \(A\) isJEE Mains 2022 Hard
- Let \(y=y(t)\) be a solution of the differential equation \(\frac{d y}{d t}+\alpha y=\gamma e^{-\beta t}\) Where, \(\alpha > 0, \beta > 0\) and \(\gamma > 0\). Then \(\operatorname{Lim}_{t \rightarrow \infty} y(t)\)JEE Mains 2023 Hard
- The probability of a man hitting a target is \(\frac{1}{10}\). The least number of shots required, so that the probability of his hitting the target at least once is greater than \(\frac{1}{4},\) isJEE Mains 2020 Hard
- \(\mathop \smallint \limits_0^\pi \sqrt {1 + 4{{\sin }^2}\frac{x}{2} - 4\sin \frac{x}{2}} \;dx = \)JEE Mains 2014 Hard
- The angles \(A, B\) and \(C\) of a triangle \(ABC\) are in \(A.P\) and \(a : b = 1 : \sqrt 3 .\) If \(c = 4\, cm,\) then the area (in sq. cm) of this triangle isJEE Mains 2019 Hard
- Let \(a_{1}=b_{1}=1, a_{n}=a_{n-1}+2\) and \(b_{n}=a_{n}+b_{n-1}\) for every natural number \(n \geq 2\). Then \(\sum_{ n =1}^{15} a _{ n } \cdot b _{ n }\) is equal to \(.........\)JEE Mains 2022 Hard