Punjab State Board PSEB 12th Class Maths Book Solutions Chapter 4 Determinants Ex 4.3 Textbook Exercise Questions and Answers.
PSEB Solutions for Class 12 Maths Chapter 4 Determinants Ex 4.3
Question 1.
Find area of the triangle with vertices at the point given in each of the following:
(i) (a, 0), (6, 0), (4, 3)
(ii) (2, 7), (1, 1), (10, 8)
(iii) (- 2, – 3), (3, 2), (- 1, – 8)
Solution.
(i) The area of the triangle with vertices (1, 0), (6, 0) and (4, 3) is given by the relation,
∆ = \(\frac{1}{2}\) \(\left|\begin{array}{lll}
1 & 0 & 1 \\
6 & 0 & 1 \\
4 & 3 & 1
\end{array}\right|\)
= \(\frac{1}{2}\) [1 (0 – 3) – 0 (6 – 4) + 1 (18 – 0)]
= \(\frac{1}{2}\) [- 3 + 18]
= \(\frac{15}{2}\) sq. unit.
(ii) The area of the triangle with vertices (2, 7), (1,1) and (10,8), is given by the relation,
∆ = \(\frac{1}{2}\) \(\left|\begin{array}{ccc}
2 & 7 & 1 \\
1 & 1 & 1 \\
10 & 8 & 1
\end{array}\right|\)
= \(\frac{1}{2}\) [2(1 – 8) – 7(1 – 10) + 1(8 – 10)]
= \(\frac{1}{2}\) [2(- 7) – 7 (- 9) + 1 (- 2)]
= \(\frac{1}{2}\) [- 14 + 63 – 2]
= \(\frac{1}{2}\) [- 16 + 63]
= \(\frac{47}{2}\) sq. unit
(iii) The area of the triangle with vertices (- 2, – 3), (3, 2) and (- 1, – 8) is given by the relation,
∆ = \(\frac{1}{2}\) \(\left|\begin{array}{ccc}
-2 & -3 & 1 \\
3 & 2 & 1 \\
-1 & -8 & 1
\end{array}\right|\)
= \(\frac{1}{2}\) [- 2(2 + 8) + 3(3 + 1) + 1(- 24 + 2)]
= \(\frac{1}{2}\) [- 2 (10) + 3(4) + 1 (- 22)]
= \(\frac{1}{2}\) [- 20 + 12 – 22]
= – \(\frac{30}{2}\)
= -15 sq. unit.
Hence, the area of the triangle is |- 15 | = 15 sq. unit.
Question 2.
Show that points A(a, b + c), B(b, c + a), C(c, a + b) are collinear.
Solution.
Area of ∆ABC is given by the relation, ∆ = \(\frac{1}{2}\) \(\left|\begin{array}{ccc}
a & b+c & 1 \\
b & c+a & 1 \\
c & a+b & 1
\end{array}\right|\)
= \(\frac{1}{2}\) \(\left|\begin{array}{ccc}
a & b+c & 1 \\
b-c & a-b & 0 \\
c-a & a-c & 0
\end{array}\right|\)
(Applying R2 → R2 – R1 and R → R3 – R1)
= \(\frac{1}{2}\) (a – b) (c – a) \(\left|\begin{array}{ccc}
a & b+c & 1 \\
-1 & 1 & 0 \\
1 & -1 & 0
\end{array}\right|\)
= \(\frac{1}{2}\) (a – b) (c – a) \(\left|\begin{array}{ccc}
a & b+c & 1 \\
-1 & 1 & 0 \\
0 & 0 & 0
\end{array}\right|\) = 0
(Applying R3 → R3 + R2)
Thus, the area of the triangle formed by points A, B and C is zero.
Hence, the points A, B and C are collinear.
Question 3.
Find the value of k, if area of triangle is 4 sq. units and vertices are
(i) (k, 0), (4, 0), (0, 2)
(ii) (- 2, 0), (0, 4), (0, k)
Solution.
(i) The area of the triangle with vertices (k, 0), (4,0) and (0, 2) is given by the relation,
∆ = \(\frac{1}{2}\) \(\left|\begin{array}{lll}
k & 0 & 1 \\
4 & 0 & 1 \\
0 & 2 & 1
\end{array}\right|\) = 4
= \(\frac{1}{2}\) [k(0 – 2) – 0 (4 – 0) + 1 (8 – 0)] = 4
∴ – k + 4 = ± 4
When – k + 4 = – 4, then k = 8
When – k + 4 = 4, then k = 0
Hence, k = 0, 8.
(ii) The area of the triangle with vertices (- 2, 0), (0, 4) and (0, k) is given by the relation,
∆ = \(\frac{1}{2}\) \(\left|\begin{array}{ccc}
-2 & 0 & 1 \\
0 & 4 & 1 \\
0 & k & 1
\end{array}\right|\) = 4
∴ k – 4 = ± 4
When k – 4 = – 4, then k = 0
When k – 4 = 4, then k = 8
Hence, k = 0, 8.
Question 4.
(i) Find the equation of the line joining (1, 2) and (3, 6) using determinants.
(ii) Find the equation of the line joining (3, 1) and (9, 3) using determinants.
Solution.
(i) LetP(x, y) be any point on the line joining points A (1, 2) and B (3, 6).
Then, the points A, B and P are collinear. Therefore, the area of triangle ABP will be zero.
∴ \(\frac{1}{2}\) \(\left|\begin{array}{lll}
1 & 2 & 1 \\
3 & 6 & 1 \\
x & y & 1
\end{array}\right|\) = 0
⇒ 6 – y – 6 + 2x + 3y – 6x = 0
⇒ 2y – 4x = 0
⇒ 2x – y = 0
Hence, the equation of the line joining the given points is 2x – y = 0.
(ii) Let P (x, y) be any point on the line joining points A (3, 1) and B (9, 3).
Then, the points A, B and P are collinear.
Therefore, the area of triangle ABP will be zero.
∴ \(\frac{1}{2}\) \(\left|\begin{array}{lll}
3 & 1 & 1 \\
9 & 3 & 1 \\
x & y & 1
\end{array}\right|\) = 0
⇒ \(\frac{1}{2}\) [3 (3 – y) – 1(9 – x) + 1 (9y – 3x)] = 0
⇒ 9 – 3y – 9 + x + 9y – 3x = 0
⇒ 6y – 2x = 0
⇒ x – 3y = 0
Hence, the equation of the line joining the given points is x – 3y = 0.
Question 5.
If area of triangle is 35 sq. units with vertices (2, – 6), (5, 4) and (k, 4). Then, k is
(A) 12
(B) – 2
(C) – 12, – 2
(D) 12, – 2
Solution.
The area of the triangle with vertices (2, – 6), (5, 4), and (k, 4) is given by the relation.
∆ = \(\frac{1}{2}\) \(\left|\begin{array}{ccc}
2 & -6 & 1 \\
5 & 4 & 1 \\
k & 4 & 1
\end{array}\right|\)
= \(\frac{1}{2}\) [2 (4 – 4) + 6 (5 – k) +1 (20 – 4k)]
= \(\frac{1}{2}\) [30 – 6k + 20 – 4k]
= \(\frac{1}{2}\) [50 – 10k]
= 25 – 5k
It is given that the area of the triangle is ± 35.
Therefore, we have
⇒ 25- 5k = ± 35
⇒ 5(5 – k) = ± 35
⇒ 5 – k = ±7
When 5 – k = – 7, then k = 5 + 7 = 12
When 5 – k = 7, then k = 5 – 7 = – 2
Hence, k = 12, – 2.
The correct answer is (D).