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Step-by-Step Solution
Step 1: Identify the Parabola and Its Focus
The given parabola is
$y^2 = 8x.$
Comparing with the standard form
$y^2 = 4ax,$
we get
$4a = 8 \implies a = 2.$
Hence, the focus of the parabola is
$(a, 0) = (2, 0).$
Step 2: Determine the Chord of Contact
Let the external point be
$(-8, 0).$
For a parabola
$y^2 = 4ax,$
the chord of contact of tangents from an external point
$(x_1, y_1)$
is given by the equation
$T = 0,$
which translates to
$yy_1 = 2a (x + x_1).$
In our case,
$a = 2,\, (x_1, y_1) = (-8, 0).$
Therefore,
$yy_1 = 4 (x + x_1).$
Substituting
$y_1 = 0,$
we get
$y \cdot 0 = 4 (x - 8)$
leading to
$x = 8.$
Step 3: Find the Coordinates of Points of Contact P and Q
The chord of contact is
$x = 8.$
Substituting
$x=8$
into the parabola
$y^2 = 8x,$
we have
$y^2 = 8(8) = 64.$
Hence,
$y = \pm 8.$
Therefore, the points of contact are
$P(8, 8)$
and
$Q(8, -8).$
Step 4: Calculate the Area of Triangle PFQ
The focus is
$F(2, 0).$
Points
$P(8, 8)$
and
$Q(8, -8).$
The base of the triangle along the x-axis is the horizontal distance between
$x = 2$
(focus) and
$x = 8$
(common x-coordinate of P and Q). Hence, base
$= 8 - 2 = 6.$
The height of the triangle is the vertical distance between
$y=8$
and
$y=-8,$
which is
$8 - (-8) = 16.$
Therefore, the area of triangle
$PFQ$
is
$ \frac{1}{2} \times \text{base} \times \text{height}
= \frac{1}{2} \times 6 \times 16
= 48
\text{ sq. units}.$
