Name: Condition for Concurrency of Three Lines
Uploaded: 2019-07-19
Duration: 10 min 26 s
Description: Well, this video beautifully explains the conditions for concurrency of three lines with some interesting examples. Come, let's learn and watch the video...!!

Question

An equilateral triangle

P Q R

is formed where

P (1, 3)

is a fixed point and

Q

is moving point on line

x = 3

. Find the locus of

R

.

Accepted Answer

Given that $P Q R$ is an equilateral triangle.

From figure, slope of $P R$ is $(θ \pm 60 °)$ .

Let the coordinates of point $R$ be $(h, k)$ and $Q$ is the variable point on the line $x = 3$ .

In $△ P S Q$ ,

$\frac{P S}{P Q} = \cos θ \Rightarrow \frac{2}{P Q} = \cos θ \Rightarrow P Q = 2 \sec θ$

Equation of a line $P R$ in parametric form, we get

$∴ \frac{h - 1}{\cos (θ \pm 60 °)} = \frac{k - 3}{\sin (θ \pm 60 °)} = 2 \sec θ$

$\Rightarrow h = 1 + 2 \cos (θ \pm 60 °) \sec θ & k = 3 + 2 \sin (θ \pm 60 °) \sec θ$

$\Rightarrow h = 1 + 2 [\cos θ \cos (60 °) \pm \sin θ \sin (60 °)] \sec θ = 1 + 2 [\frac{1}{2} \pm \frac{\sqrt{3}}{2} \tan θ] \dots (1)$

and $k = 3 + 2 [\sin θ \cos (60 °) \pm \cos θ \sin (60 °)] \sec θ = 3 + 2 [\frac{1}{2} \tan θ \pm \frac{\sqrt{3}}{2}] \dots (2)$

To eliminate $θ$ ,

$\frac{h - 2}{\sqrt{3}} = \pm \tan θ$ and $k - 3 \pm \sqrt{3} = \tan θ$

$\Rightarrow \frac{h - 2}{\sqrt{3}} = \pm (k - 3 \pm \sqrt{3})$

$\Rightarrow h - 2 = \pm \sqrt{3} (k - 3 \pm \sqrt{3})$

Replace $h \to x, k \to y$ , we get locus of a point $R$ is

$x - 2 = \pm \sqrt{3} (y - 3 \pm \sqrt{3})$

An equilateral triangle $P Q R$ is formed where $P (1, 3)$ is a fixed point and $Q$ is moving point on line $x = 3$ . Find the locus of $R$ .

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

An equilateral triangle PQR is formed where P(1,3) is a fixed point and Q is moving point on line x=3. Find the locus of R.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

An equilateral triangle $P Q R$ is formed where $P (1, 3)$ is a fixed point and $Q$ is moving point on line $x = 3$ . Find the locus of $R$ .