Tangents drawn from the point $(-8, 0)$ to the parabola $y^2 = 8x$ touch the parabola at $P$ and $Q$ . If $F$ is the focus of the parabola, then the area of the triangle $PFQ$ (in s units) is equal to :

Equation of tangent for parabola $y^{2} = 8x$ $y = mx+\frac{2}{m}$ tangent passing through $\left(-8, 0\right)$ $0 = -8m+\frac{2}{m}$ $m^{2} = \frac{1}{4}$ $m = \pm \frac{1}{2}$ for point $P\left(\frac{a}{m^{2}}, \frac{2a}{m}\right) = \left(\frac{2}{1/2}, \frac{4}{1/2}\right) = \left(8, 8\right)$ $Q\left(\frac{2}{1/4}, \frac{4}{-1/2}\right) = \left(8, -8\right)$ Area of $\Delta PFQ = \frac{1}{2}\times16\times6 = 48$ sunits.

Tangents drawn from the point $(-8, 0)$ to the par

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Parabola

Parabola is defined as the locus of points equidistant from a fixed point (called focus) and a fixed-line (called directrix).

Standard Equation of a Parabola

For horizontal parabola

Let us consider
Origin (0,0) as the parabola's vertex A,

Two equidistant points S(a,0) as focus, and Z(- a,0) as a directrix point,
P(x,y) as the moving point.

Let us now draw SZ perpendicular from S to the directrix. Then, SZ will be the axis of the parabola.
The centre point of SZ i.e. A will now lie on the locus of P, i.e. AS = AZ.
The x-axis will be along the line AS, and the y-axis will be along the perpendicular to AS at A, as in the figure.
By definition PM = PS

=> MP² = PS²

So, (a + x)² = (x - a)² + y².
Hence, we can get the equation of horizontal parabola as y² = 4ax.

For vertical parabola

Let us consider
Origin (0,0) as the parabola's vertex A

Two equidistant points, S(0,b) as focus and Z(0, -b) as a directrix point
P(x,y) as any moving point

Let us now draw a perpendicular SZ from S to the directrix.
Then SZ will be the axis of the parabola. Now, the midpoint of SZ i.e. A, will lie on P’s locus i.e. AS=AZ.
The y-axis will be along the line AS, and the x-axis will be perpendicular to AS at A, as shown in the figure.
By definition PM = PS

=> MP² = PS²

So, (b + y)² = (y - b)² + x²

As a result, the vertical parabola equation is x²= 4by.