Question

Let $A$ lies on $3x-4y+1=0,$ $B$ lies on $4x+3y-7=0$ and $C$ is $\left(- 2 , 5\right).$ If $ABCD$ is a rhombus, then the locus of $D$ is a conic whose length of the latus rectum is equal to

• A. $10$ units
• B. $15$ units
• C. $5$ units
• D. $20$ units

Answer 1

Answer: (A)

Let, $D\left(h , k\right)$
$C\left(- 2 , 5\right)$ lies on $4x+3y-7=0$
$\Rightarrow BC$ is perpendicular to $3x-4y+1=0$
$\Rightarrow DA$ is perpendicular to $3x-4y+1=0$
$\Rightarrow $ Because $DA=DC$ so, the distance of $D$ from $3x-4y+1=0$ is equal to $DC$
$\Rightarrow \sqrt{\left(h + 2\right)^{2} + \left(k - 5\right)^{2}}=\left|\frac{3 h - 4 k + 1}{5}\right|$
$\Rightarrow 25\left(\left(x + 2\right)^{2} + \left(y - 5\right)^{2}\right)=\left(3 x - 4 y + 1\right)^{2}$
Which represents parabola with focus $\left(- 2 , 5\right)$ and directrix is $3x-4y+1=0$
$\Rightarrow $ Length of the latus rectum is $2\left|\frac{- 6 - 20 + 1}{5}\right|=10$ units