P.T the area of the triangle inscribed in the parabola y2 = 4ax is 18ay1−y2y2−y3y3−y1 sqare unit where y1, y2, y3 are the ordinates of its vertices.

Given parabola y2 = 4axLet A(x1, y1),B(x2 , y2 ),C(x3, y3 ) be there points on the parabola y2 = 4axy12=4ax1 ⇒x1=y124a,x2=y224a and x3=y324a∴Ay124a,y1,By224a,y2,Cy324a,y3 are the points on parabola y2 = 4axArea of Triangle =Absolute value of12x1−x2 y1−y2x2−x3 y2−y3=Absolute value of 12y12−y224ay1−y2y22−y324ay2−y3=Absolute value of1214ay12−y22y1−y2y22−y32y2−y3=Absolute value of18ay1−y2y2−y3y1+y21y2+y31=Absolute value of18ay1−y2y2−y3y1−y3=18ay1−y2y2−y3y3−y1 sq. units

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P.T the area of the triangle inscribed in the parabola y² = 4ax is $\frac{1}{8 a} |(y_{1} - y_{2}) (y_{2} - y_{3}) (y_{3} - y_{1})|$ sqare unit
where y₁, y₂, y₃ are the ordinates of its vertices.

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Detailed Solution

Given parabola y² = 4ax
Let A(x₁, y₁),B(x₂ , y₂ ),C(x₃, y₃ ) be there points on the parabola y² = 4ax
$y_{1}^{2} = 4 {ax}_{1} \Rightarrow x_{1} = \frac{y_{1}^{2}}{4 a}, x_{2} = \frac{y_{2}^{2}}{4 a} and x_{3} = \frac{y_{3}^{2}}{4 a}$
$∴ A (\frac{y_{1}^{2}}{4 a}, y_{1}), B (\frac{y_{2}^{2}}{4 a}, y_{2}), C (\frac{y_{3}^{2}}{4 a}, y_{3})$ are the points on parabola y² = 4ax
Area of Triangle $= A b s o l u t e v a l u e o f \frac{1}{2} |\begin{array}{ll} x_{1} - x_{2} y_{1} - y_{2} \\ x_{2} - x_{3} y_{2} - y_{3} \end{array}|$
$\begin{array}{l} = A b s o l u t e v a l u e o f \frac{1}{2} |\begin{array}{ll} \frac{y_{1}^{2} - y_{2}^{2}}{4 a} & y_{1} - y_{2} \\ \frac{y_{2}^{2} - y_{3}^{2}}{4 a} & y_{2} - y_{3} \end{array}| \\ = A b s o l u t e v a l u e o f \frac{1}{2} \frac{1}{4 a} |\begin{array}{ll} y_{1}^{2} - y_{2}^{2} & y_{1} - y_{2} \\ y_{2}^{2} - y_{3}^{2} & y_{2} - y_{3} \end{array}| \\ = A b s o l u t e v a l u e o f \frac{1}{8 a} (y_{1} - y_{2}) (y_{2} - y_{3}) |\begin{array}{ll} y_{1} + y_{2} & 1 \\ y_{2} + y_{3} & 1 \end{array}| \\ = A b s o l u t e v a l u e o f \frac{1}{8 a} (y_{1} - y_{2}) (y_{2} - y_{3}) |y_{1} - y_{3}| \\ = \frac{1}{8 a} |(y_{1} - y_{2}) (y_{2} - y_{3}) (y_{3} - y_{1})| sq. units \end{array}$