Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • All lines represented by the equation<img alt="\mathrm{ (2 \cos \theta+3 \sin \theta) x+(3 \cos \theta-5 \sin \theta) y=5 \cos \theta-2 \sin \theta }" src="https://entrancecorner.oncodecogs.com/gif

All lines represented by the equation\mathrm{ (2 \cos \theta+3 \sin \theta) x+(3 \cos \theta-5 \sin \theta) y=5 \cos \theta-2 \sin \theta }(1)

pass through a fixed point for all \mathrm{ \theta }. What are the coordinates of this fixed point and its reflection in the line \mathrm{ x+y=\sqrt{2}, } respectively?

Option: 1

\mathrm{(1,1) and (\sqrt{2}, \sqrt{2})}

 


Option: 2

\mathrm{(1,1) and (\sqrt{2}+1, \sqrt{2}+1)}
 


Option: 3

\mathrm{(1,1) and (\sqrt{2}-1, \sqrt{2}-1)}
 


Option: 4

\mathrm{(\sqrt{2}, \sqrt{2}) and (1,1)}


Answers (1)

best_answer

The equation (1) can be expressed as

\mathrm{ (2 x+3 y-5) \cos \theta+(3 x-5 y+2) \sin \theta=0 }

\mathrm{ \text { or } \quad(2 x+3 y-5)+\tan \theta(3 x-5 y+2)=0}

It is clear that these lines will pass through the point of intersection of the lines

\mathrm{ 2 x+3 y-5=0 }                (3)

\mathrm{ 3 x-5 y+2=0}

For all values of \mathrm{\theta}. Solving equations (3) and (4) we get the coordinates of the required fixed points as \mathrm{\mathrm{P}(1, l)}.

Let \mathrm{Q(\alpha, \beta)} be the reflection of \mathrm{P(1,1)} in the line \mathrm{x+y=\sqrt{2}}. The line \mathrm{P Q} is perpendicular to the line \mathrm{x+y=\sqrt{2}} and the mid point of segment \mathrm{P Q} lies on the line
\mathrm{x+y=\sqrt{2}}. Thus

\mathrm{ \left(\frac{\beta-1}{\alpha-1}\right)(-1)=-1 \quad \Rightarrow \quad \beta=\alpha }

\mathrm{and \frac{\beta+1}{2}+\frac{\alpha+1}{2}=\sqrt{2} \Rightarrow \beta=\alpha=\sqrt{2}-1 }

Hence coordinates of the reflection \mathrm{Q} of \mathrm{P} in the line \mathrm{x+y=\sqrt{2}} are

\mathrm{ Q \equiv(\sqrt{2}-1, \sqrt{2}-1) }

Hence option 3 is correct.
 

Posted by

Sanket Gandhi

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2026 April Exam LIVE: NTA Day 2 entrance exam tomorrow; shift timings, answer key
anam-khan

JEE Main 2026 April 2 Shift 2 Analysis: Physics easy, maths tricky, say students
anam-khan

JEE Main 2026 April 2 Shift 1 Analysis: Paper 'easy to moderate'; students find maths challenging

Latest Question