Question

If point $p$ ,shown in figure, is moved on hyperbola, then $F_{1}P-F_{2}P$ will

• A. always positive
• B. always negative
• C. always same
• D. changes it's sign when point cross the $x$ - axis.

Answer 1

Answer: (A), (C)

always positive
always same

Let $Z$ and $Z^{\prime }$ are the two directrix of this Hyperbola right hand-side and left hand-side respectively.

As we know Directrix of Hyperbola is always parallel to Conjugate axis or minor axis of hyperbola. The distance of directrix of hyperbola from the center is $\frac{a}{e}$ , both sides.

So equation of $Z$ is $x=\frac{a}{e}$ and equation of $Z^{\prime }$ is $x=-\frac{a}{e}$

Now if a perpendicular is drawn from point $P$ to Both Directrix, let it meet $Z$ at $M$ and $Z^{\prime }$ at $M^{\prime }$.

Then according to definition of Hyperbola $F_{2}P=ePM$ and $F_{1}P=eP{M}^{\prime }$

Now let the coordinates of point $P$ are $(x,y)$

Then $PM=x-\frac{a}{e}$

and $P{M}^{\prime }=x-(-\frac{a}{e})=x+\frac{a}{e}$

We know that $F_{2}P=ePM=e(x-\frac{a}{e})$

$\to$ $F_{2}P=ex-a$ ......$\left(1\right)$

Similiarly $F_{1}P=eP{M}^{\prime }=e(x+\frac{a}{e})$

$\to$ $F_{1}P=ex+a$ .....$\left(2\right)$

Subtracting eq. $\left(1\right)$ from eq. $\left(2\right)$ We get, $F_{1}P-F_{2}P=ex+a-(ex-a)$

$\to$ $F_{1}P-F_{2}P=2a$

As for a Hyperbola $a>0$ so the value of $F_{1}P-F_{2}P$ is always Positive and also Constant as $2a$ is a constant value.

Hence option $A$ and $C$ are correct.