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

A variable line cuts the line $2 y = x - 2$ and $2 y = - x + 2$ in points $A$ and $B$ respectively. If $A$ lies in first quadrant, $B$ lies in $4^{th}$ quadrant and area of $∆ A O B$ is $4$ sq. units, then find locus of mid point of $A B$ .

Accepted Answer

Since $A$ lies on $2 y = x - 2$ , let $A (2 a + 2, a)$ .

Since $B$ lies on $2 y = - x + 2$ , let $B (2 - 2 b, b)$ .

Let mid point of $A B$ be $P (h, k) \equiv (\frac{2 a + 2 + 2 - 2 b}{2}, \frac{a + b}{2})$ i.e. $(a - b + 2, \frac{a + b}{2})$ .

From $h = a - b + 2 & k = \frac{a + b}{2}$ , we get $a = \frac{h + 2 k - 2}{2}, b = \frac{- h + 2 k + 2}{2} . . (1)$

Now, area of $∆ A O B = \frac{1}{2} ||\begin{matrix} 0 & 0 & 1 \\ 2 a + 2 & a & 1 \\ 2 - 2 b & b & 1 \end{matrix}||$ sq. units i.e. $\frac{1}{2} |(0 - 0 + 1 (2 a + 2) b - (2 - 2 b) a)|$ sq. units i.e. $|2 a b + b - a|$ sq. units which is given to be $4$ sq. units

$\Rightarrow {(2 a b + b - a)}^{2} = 16$

$\Rightarrow {(2 (\frac{h + 2 k - 2}{2}) (\frac{- h + 2 k + 2}{2}) + \frac{- h + 2 k + 2}{2} - \frac{h + 2 k - 2}{2})}^{2} = 16$ (From $(1)$ )

$\Rightarrow {(2 k)}^{2} - {(h - 2)}^{2} - 2 h + 4 = 64$

$\Rightarrow 4 k^{2} - {(h - 1)}^{2} = 63$

So, required locus is $4 y^{2} - {(x - 1)}^{2} = 63$

A variable line cuts the line $2 y = x - 2$ and $2 y = - x + 2$ in points $A$ and $B$ respectively. If $A$ lies in first quadrant, $B$ lies in $4^{th}$ quadrant and area of $∆ A O B$ is $4$ sq. units, then find locus of mid point of $A B$ .

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

A variable line cuts the line 2y=x-2 and 2y=-x+2 in points A and B respectively. If A lies in first quadrant, B lies in 4th quadrant and area of ∆AOB is 4 sq. units, then find locus of mid point of AB.

Important Questions on Point and Straight Line

Learn and Prepare for any exam you want !

A variable line cuts the line $2 y = x - 2$ and $2 y = - x + 2$ in points $A$ and $B$ respectively. If $A$ lies in first quadrant, $B$ lies in $4^{th}$ quadrant and area of $∆ A O B$ is $4$ sq. units, then find locus of mid point of $A B$ .