Question

If the vertex and the focus of parabola are (3,6) and (4,5)then the equation of its directrix is:

• A. x-y+7=0
• B. x-y+5=0
• C. x-y+9=0
• D. x-y+3=0

Answer 1

The correct option is B x-y+5=0

Given:-

Vertex $\equiv (3,6)$

Focus $\equiv (4,5)$

Equation of axis of symmetry-

$(y-6)=\frac{5-6}{4-3}(x-3)$

$y-6=-x+3$

$\Rightarrow x+y-9=0$

$\Rightarrow$ slope of axis of symmetry $=-1$

Therefore, slope of directrix $=1$

As we know that in a parabola, vertex is the mid-point of of focus and the point of intersection of directrix and axis of symmetry.

Now,

Let co-ordinate of the point of intersection of direction and axis of symmetriy be $(a,b)$ then

$\frac{a+4}{2}=3$

$\Rightarrow a+4=6$

$\Rightarrow a=6-4=2$

$\frac{b+5}{2}=6$

$\Rightarrow b+5=12$

$\Rightarrow b=12-5=7$

Thus the point of intersection is $(2,7)$.

Therefore,

Equation of directrix will be-

$(y-7)=1(x-2)$

$\Rightarrow x-y+5=0$

Hence the equation of directrix is $x-y+5=0$.

Hence the correct answer is $\left(B\right)x-y+5=0$.