Prove that the area of the triangle inscribed in the parabola y 2 = 4 a x is 1 8 a y 1 - y 2 y 2 - y 3 y 3 - y 1 sq units where y 1 , y 2 , y 3 are the ordinates of its vertices.

Given equation of the parabola y 2 = 4 a x Let P x 1 , y 1 = a t 1 2 , 2 a t 1 , Q x 2 , y 2 = a t 2 2 , 2 a t 2 , R x 3 , y 3 = a t 3 2 , 2 a t 3 be the vertices of the △ P Q R Area of △ P Q R = 1 2 a t 1 2 2 a t 1 1 a t 2 2 2 a t 2 1 a t 3 2 2 a t 3 1 = 1 2 a t 1 2 2 a t 2 - 2 a t 3 - 2 a t 1 a t 2 2 - a t 3 2 + 1 2 a 2 t 3 t 2 2 - 2 a 2 t 2 t 3 2 = 2 a 2 2 t 1 2 t 2 - t 3 - t 1 t 2 - t 3 t 2 + t 3 + t 2 t 3 t 2 - t 3 = a 2 t 2 - t 3 t 1 2 - t 1 t 2 - t 1 t 3 + t 2 t 3 = a 2 t 2 - t 3 t 1 t 1 - t 2 - t 3 t 1 - t 2 = a 2 t 2 - t 3 t 1 - t 3 t 1 - t 2 = a 2 8 a 3 2 a t 1 - 2 a t 2 2 a t 2 - 2 a t 3 2 a t 3 - 2 a t 1 = 1 8 a y 1 - y 2 y 2 - y 3 y 3 - y 1 sq units Hence, proved.

E M B I B E

Intermediate Second Year Mathematics Paper 2B>Coordinate Geometry>Chapter 3 - Parabola>Exercise 3(a)

Exercise 3(a)

Author:Andhra Pradesh Board

Andhra Pradesh Board Mathematics Solutions for Exercise 3(a)

Simple step-by-step solutions to Exercise 3(a) questions of Parabola from Intermediate Second Year Mathematics Paper 2B. Also get 3D topic explainers, cheat sheets, and unlimited doubts solving on EMBIBE.

Questions from Exercise 3(a) with Hints & Solutions

EASY

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2B>Chapter 3 - Parabola>Exercise 3(a)>Q I

Find the position (interior or exterior or on) of the point $(6, - 6)$ with respect to parabola $y^{2} = 6 x$ .

EASY

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2B>Chapter 3 - Parabola>Exercise 3(a)>Q I

Find the position (interior or exterior or on) of the point $(0, 1)$ with respect to parabola $y^{2} = 6 x$ .

EASY

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2B>Chapter 3 - Parabola>Exercise 3(a)>Q I

Find the position (interior or exterior or on) of the point $(2, 3)$ with respect to parabola $y^{2} = 6 x$ .

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2B>Chapter 3 - Parabola>Exercise 3(a)>Q II

Find the equation of the parabola whose axis is parallel to $y -$ axis and which passes through these points $(4, 5), (- 2, 11)$ and $(- 4, 21)$ .

HARD

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2B>Chapter 3 - Parabola>Exercise 3(a)>Q III

Find the equation of the parabola whose focus is $(- 2, 3)$ and directrix is the line $2 x + 3 y - 4 = 0$ . Also find the length of the latus rectum and equation of the axis of the parabola.

MEDIUM

12th Andhra Pradesh Board

IMPORTANT

Intermediate Second Year Mathematics Paper 2B>Chapter 3 - Parabola>Exercise 3(a)>Q III

Prove that the area of the triangle inscribed in the parabola $y^{2} = 4 a x$ is $\frac{1}{8 a} |(y_{1} - y_{2}) (y_{2} - y_{3}) (y_{3} - y_{1})|$ sq units where $y_{1}, y_{2}, y_{3}$ are the ordinates of its vertices.