RD Chapter 23 The Straight Lines Ex 23.14 Solutions
Question - 1 : - Find the values of α so that the point P(α 2, α) lies inside or on the triangle formed by the lines x – 5y + 6 = 0, x – 3y + 2 = 0 and x – 2y – 3 = 0.
Answer - 1 : -
Given:
x – 5y + 6 = 0, x – 3y+ 2 = 0 and x – 2y – 3 = 0 forming a triangle and point P(α2, α)lies inside or on the triangle
Let ABC be thetriangle of sides AB, BC and CA whose equations are x − 5y + 6 = 0,x − 3y + 2 = 0 and x − 2y − 3 = 0, respectively.
On solving theequations, we get A (9, 3), B (4, 2) and C (13, 5) as the coordinates of thevertices.
It is given that pointP (α2, α) lies either inside or on the triangle. The threeconditions are given below.
(i) A and P must lieon the same side of BC.
(ii) B and P must lieon the same side of AC.
(iii) C and P must lieon the same side of AB.
If A and P lie on thesame side of BC, then
(9 – 9 + 2)(α2 –3α + 2) ≥0
(α – 2)(α –1) ≥ 0
α ∈ (- ∞, 1 ] ∪ [ 2, ∞) … (1)
If B and P lie on thesame side of AC, then
(4 – 4 – 3) (α2 –2α – 3) ≥ 0
(α – 3)(α +1) ≤ 0
α ∈ [- 1, 3] … (2)
If C and P lie on thesame side of AB, then
(13 – 25 + 6)(α2 –5α + 6) ≥0
(α – 3)(α –2) ≤ 0
α ∈ [ 2, 3] … (3)
From equations (1),(2) and (3), we get
α∈ [2, 3]
∴ α∈ [2, 3]
Question - 2 : - Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y – 4 = 0, 3x – 7y – 8 = 0 and 4x – y – 31 = 0.
Answer - 2 : -
Given:
x + y – 4 = 0, 3x – 7y– 8 = 0 and 4x – y – 31 = 0 forming a triangle and point (a, 2)is an interiorpoint of the triangle
Let ABC be thetriangle of sides AB, BC and CA whose equations are x + y − 4 = 0,3x − 7y − 8 = 0 and 4x − y − 31 = 0,respectively.
On solving them, weget A (7, – 3), B (18/5, 2/5) and C (209/25, 61/25) as thecoordinates of the vertices.
Let P (a, 2) be thegiven point.
It is given that pointP (a, 2) lies inside the triangle. So, we have the following:
(i) A and P must lieon the same side of BC.
(ii) B and P must lieon the same side of AC.
(iii) C and P must lieon the same side of AB.
Thus, if A and P lieon the same side of BC, then
21 + 21 – 8 – 3a – 14– 8 > 0
a > 22/3 … (1)
From (1), (2) and (3),we get:
A ∈ (22/3, 33/4)
∴ A ∈ (22/3, 33/4)
Question - 3 : - Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations x + y – 4 = 0, 3x – 7y + 8 = 0, 4x – y – 31 = 0.
Answer - 3 : -
Given:
x + y – 4 = 0, 3x – 7y+ 8 = 0, 4x – y – 31 = 0 forming a triangle and point (-3, 2)
Let ABC be thetriangle of sides AB, BC and CA, whose equations x + y − 4 = 0,3x − 7y + 8 = 0 and 4x − y − 31 = 0,respectively.
On solving them, weget A (7, – 3), B (2, 2) and C (9, 5) as the coordinates of the vertices.
Let P (− 3, 2) bethe given point.
The given point P(− 3, 2) will lie inside the triangle ABC, if
(i) A and P lies onthe same side of BC
(ii) B and P lies onthe same side of AC
(iii) C and P lies onthe same side of AB
Thus, if A and P lieon the same side of BC, then
21 + 21 + 8 – 9 – 14 +8 > 0
50 × – 15 > 0
-750 > 0,
This is false
∴ The point (−3, 2)lies outside triangle ABC.