Question

Consider the conic $x^2+4y-6x+k=0$ & $L\Rightarrow y+1=0$ be its directrix

On the basis of above information answer the following question:

The focus of the parabola is

• A. $(-3,3)$
• B. $(-3,-3)$
• C. $(3,-3)$
• D. None of these

Answer 1

Answer: (C)

$(3,-3)$

$x^2+4y-6x+k=0$

$⟹(x-3{)}^2=-4(y-(\frac{k-9}{4}\left)\right)$

So, length of latus rectum $=4a=4$

$\therefore a=1$

Distance between vertex and directrix=a

$\therefore (-1-(\frac{k-9}{4}\left)\right)=1$

$\therefore k=1$

So, co-ordinates of vertex are $(3,\frac{k-9}{4})=(3,-2)$

Therefore, co-ordinates of focus are $(3,-2-a)=(3,-3)$

So, answer is option (C).