If two parallel chords of a circle,having dia | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The diameter of the circle is $4 \, 	ext{units}$,so the radius $r = 2 \, 	ext{units}$.Let the angles subtended by the chords at the centre be $2

Vedclass · Accepted Answer

$\frac{8}{\sqrt{7}}$

Vedclass · Answer

(B) The diameter of the circle is $4 \, 	ext{units}$,so the radius $r = 2 \, 	ext{units}$.Let the angles subtended by the chords at the centre be $2	heta$ and $2\phi$.Given $2	heta = \cos^{-1}(1/7) \Rightarrow \cos(2	heta) = 1/7$.Using the formula $\cos(2	heta) = 2\cos^2	heta - 1$,we have $2\cos^2	heta - 1 = 1/7$ $\Rightarrow 2\cos^2	heta = 8/7$ $\Rightarrow \cos^2	heta = 4/7$ $\Rightarrow \cos	heta = 2/\sqrt{7}$.The distance of the first chord from the centre is $d_1 = r \cos	heta = 2 	imes (2/\sqrt{7}) = 4/\sqrt{7}$.Given $2\phi = \sec^{-1}(7)$ $\Rightarrow \sec(2\phi) = 7$ $\Rightarrow \cos(2\phi) = 1/7$.Using the formula $\cos(2\phi) = 2\cos^2\phi - 1$,we have $2\cos^2\phi - 1 = 1/7$ $\Rightarrow 2\cos^2\phi = 8/7$ $\Rightarrow \cos^2\phi = 4/7$ $\Rightarrow \cos\phi = 2/\sqrt{7}$.The distance of the second chord from the centre is $d_2 = r \cos\phi = 2 	imes (2/\sqrt{7}) = 4/\sqrt{7}$.Since the chords lie on opposite sides of the centre,the total distance between them is $d_1 + d_2 = 4/\sqrt{7} + 4/\sqrt{7} = 8/\sqrt{7}$.

If two parallel chords of a circle,having diameter $4 \, \text{units}$,lie on the opposite sides of the centre and subtend angles $\cos^{-1}\left(\frac{1}{7}\right)$ and $\sec^{-1}(7)$ at the centre respectively,then the distance between these chords is:

Similar Questions

The line $y=mx+c$ intercepts the circle $x^2+y^2=r^2$ in two distinct points,if

Two circles $x^2 + y^2 = ax$ and $x^2 + y^2 = c^2$ touch each other if:

If the circles $x^2 + y^2 = a^2$ and $x^2 + y^2 - 2gx + g^2 - b^2 = 0$ touch each other externally,then:

If the points $P(3, 1)$ and $Q(6, 5)$ form a triangle with a third point $R(x, y)$ such that the area of $\Delta PQR$ is $6$ square units and $\angle PRQ = \frac{\pi}{2}$,then the number of possible positions for point $R$ is:

The area (in square units) of the circle which touches the lines $4x + 3y = 15$ and $4x + 3y = 5$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module