RD Chapter 6 Determinants Ex 6.3 Solutions
Question - 1 : - Find the area of the triangle with vertices at the points:
(i) (3, 8), (-4, 2) and (5, -1)
(ii) (2, 7), (1, 1) and (10, 8)
(iii) (-1, -8), (-2, -3) and (3, 2)
(iv) (0, 0), (6, 0) and (4, 3)
Answer - 1 : -
(i) Given (3, 8), (-4,2) and (5, -1) are the vertices of the triangle.
We know that, ifvertices of a triangle are (x1, y1), (x2, y2)and (x3, y3), then the area of the triangle is given by:
(ii) Given (2, 7), (1,1) and (10, 8) are the vertices of the triangle.
We know that ifvertices of a triangle are (x1, y1), (x2, y2)and (x3, y3), then the area of the triangle is given by:
(iii) Given (-1, -8),(-2, -3) and (3, 2) are the vertices of the triangle.
We know that ifvertices of a triangle are (x1, y1), (x2, y2)and (x3, y3), then the area of the triangle is given by:
As we know area cannotbe negative. Therefore, 15 square unit is the area
Thus area of triangleis 15 square units
(iv) Given (-1, -8),(-2, -3) and (3, 2) are the vertices of the triangle.
We know that ifvertices of a triangle are (x1, y1), (x2, y2)and (x3, y3), then the area of the triangle is given by:
Question - 2 : - Using the determinants show that the following points are collinear:
(i) (5, 5), (-5, 1) and (10, 7)
(ii) (1, -1), (2, 1) and (10, 8)
(iii) (3, -2), (8, 8) and (5, 2)
(iv) (2, 3), (-1, -2) and (5, 8)
Answer - 2 : -
(i) Given (5, 5), (-5,1) and (10, 7)
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by
(ii) Given (1, -1),(2, 1) and (10, 8)
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
(iii) Given (3, -2),(8, 8) and (5, 2)
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
Now, by substitutinggiven value in above formula
Since, Area oftriangle is zero
Hence, points arecollinear.
(iv) Given (2, 3),(-1, -2) and (5, 8)
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
Question - 3 : - If the points (a, 0), (0, b) and (1, 1) are collinear, prove that a + b = ab
Answer - 3 : -
Given (a, 0), (0, b)and (1, 1) are collinear
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
⇒
⇒ a + b = ab
Hence Proved
Question - 4 : - Using the determinants prove that the points (a, b), (a’, b’) and (a – a’, b – b) are collinear if a b’ = a’ b.
Answer - 4 : -
Given (a, b), (a’, b’)and (a – a’, b – b) are collinear
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
⇒ a b’ = a’ b
Hence, the proof.
Question - 5 : - Find the value of λ so that the points (1, -5), (-4, 5) and (λ, 7) arecollinear.
Answer - 5 : -
Given (1, -5), (-4, 5)and (λ, 7) are collinear
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
⇒ – 50 – 10λ = 0
⇒ λ = – 5
Question - 6 : - Find the value of x if the area of ∆ is 35 square cms with vertices (x,4), (2, -6) and (5, 4).
Answer - 6 : -
Given (x, 4), (2, -6)and (5, 4) are the vertices of a triangle.
We have the conditionthat three points to be collinear, the area of the triangle formed by thesepoints will be zero. Now, we know that, vertices of a triangle are (x1,y1), (x2, y2) and (x3, y3),then the area of the triangle is given by,
⇒ [x (– 10) – 4(–3) + 1(8 – 30)] = ± 70
⇒ [– 10x + 12 +38] = ± 70
⇒ ±70 = – 10x + 50
Taking positive sign,we get
⇒ + 70 = – 10x +50
⇒ 10x = – 20
⇒ x = – 2
Taking –negative sign,we get
⇒ – 70 = – 10x +50
⇒ 10x = 120
⇒ x = 12
Thus x = – 2, 12
Question - 7 : - 
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