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)

Telangana Board Solutions for Exercise 1: Exercise 3(a)

Author:Telangana Board

Telangana Board Mathematics Solutions for Exercise - Telangana Board Solutions for Exercise 1: Exercise 3(a)

Attempt the practice questions from Exercise 1: Exercise 3(a) with hints and solutions to strengthen your understanding. Intermediate Second Year Mathematics Paper 2B solutions are prepared by Experienced Embibe Experts.

Questions from Telangana Board Solutions for Exercise 1: Exercise 3(a) with Hints & Solutions

EASY

12th Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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 Telangana 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.