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 $DC$
$\Rightarrow \sqrt{{\left(h+2\right)}^2+{\left(k-5\right)}^2}=\left|\frac{3h-4k+1}{5}\right|$
$\Rightarrow 25\left({\left(x+2\right)}^2+{\left(y-5\right)}^2\right)={\left(3x-4y+1\right)}^2$
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