Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then is equal

Question

Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then ${(Q R)}^{2}$ is equal to _________. [2025]

Smart Achievers · Accepted Answer

(28)

Let $∠ P R N = θ$

$\Rightarrow P R = \cos e c θ and P Q = 4 \sec (30 ° + θ)$

For equilateral $△$ PQR.

Let PR = PQ

$\Rightarrow cosec θ = 4 \sec (θ + 30 °)$

$\Rightarrow \cos (θ + 30 °) = 4 \sin θ$

$\Rightarrow \frac{\sqrt{3}}{2} \cos θ - \frac{1}{2} \sin θ = 4 \sin θ$

$\Rightarrow \tan θ = \frac{1}{3 \sqrt{3}}$

$\Rightarrow cosec θ = \sqrt{28}$

Now, $Q R^{2} = d^{2} = {cosec}^{2} θ = 28$ .

Solution

Q.
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then ${(Q R)}^{2}$ is equal to _________. [2025]

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Solution

Q. Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then (QR)2 is equal to _________. [2025]

Ans. (28) Let ∠PRN=θ ⇒ PR=cosec θ and PQ=4 sec(30°+θ) For equilateral △PQR. Let PR = PQ ⇒ cosec θ=4 sec(θ+30°) ⇒ cos (θ+30°)=4 sin θ ⇒ 32cos θ–12sin θ=4 sin θ ⇒ tan θ=133 ⇒ cosecθ=28 Now, QR2=d2=cosec2θ=28.

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Q.
Let the distance between two parallel lines be 5 units and a point P lie between the lines at a unit distance from one of them. An equilateral triangle PQR is formed such that Q lies on one of the parallel lines, while R lies on the other. Then ${(Q R)}^{2}$ is equal to _________. [2025]