The locus of the mid points of the chords of the circle C 1 : ( x - 4 ) 2 + ( y - 5 ) 2 = 4 which subtend an angle i at the centre of the circle C 1 , is a circle of radius r i . If 1 = 3 , 3 = 2 3 and r 1 2 = r 2 2 + r 3 2 , then 2 is equal to

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
The locus of the mid points of the chords of the circle $C_{1} : {(x - 4)}^{2} + {(y - 5)}^{2} = 4$ which subtend an angle $θ_{i}$ at the centre of the circle $C_{1}$ , is a circle of radius $r_{i}$ . If $θ_{1} = \frac{π}{3}, θ_{3} = \frac{2 π}{3}$ and $r_{1}^{2} = r_{2}^{2} + r_{3}^{2},$ then $θ_{2}$ is equal to [2023]

1 $\frac{π}{2}$

2 $\frac{π}{4}$

3 $\frac{π}{6}$

4 $\frac{0 - π}{4}$

Ans.
(1)

If a chord of circle of radius $R$ subtends angle $θ_{i}$ at the centre $C_{1},$ then locus of the end point of this chord in a circle of radius

$r_{i} = R \cos (\frac{θ_{i}}{2})$

Given, $r_{1}^{2} = r_{2}^{2} + r_{3}^{2}$

$\Rightarrow \cos^{2} \frac{θ_{1}}{2} = \cos^{2} \frac{θ_{2}}{2} + \cos^{2} \frac{θ_{3}}{2}$

$\Rightarrow \cos^{2} \frac{π}{6} = \cos^{2} \frac{θ_{2}}{2} + \cos^{2} \frac{π}{3}$

$\Rightarrow \frac{3}{4} = \cos^{2} \frac{θ_{2}}{2} + \frac{1}{4}$

$\Rightarrow \cos^{2} \frac{θ_{2}}{2} = \frac{1}{2} \Rightarrow \cos \frac{θ_{2}}{2} = \frac{1}{\sqrt{2}} \Rightarrow \frac{θ_{2}}{2} = \frac{π}{4} \Rightarrow θ_{2} = \frac{π}{2}$

Want Us to Email you About Special Offers & Updates?