Question

The least positive value of $t$ so that the lines $x=t+\alpha ,y+16=0$ and $y=\alpha x$ are concurrent is

• A. $2$
• B. $4$
• C. $16$
• D. $8$

Answer 1

The correct option is D $8$

The given equations,
$x=t+\alpha ...\left(1\right)$
$y+16=0...\left(2\right)$
$y=\alpha x...\left(3\right)$

Since, the lines are concurrent, they intersect at a point.
From equation $\left(2\right)$ and $\left(3\right)$,
$x=-\frac{16}{\alpha }$
Substitute the value in equation $\left(1\right)$
$t=-(\alpha +\frac{16}{\alpha })$
The minimum value of $t$ is obtained when $\alpha =-4$.
Thus, the least value of $t$ is $8$.