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Q5:
If area of triangle is 35 sq units with vertices $(2, – 6), (5, 4)$ and $(k, 4)$. Then $k$ is
Solution :
Step 1: Establishing the Coordinate Geometry Formula
The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by the determinant formula:
$\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
[By the Determinant Definition of Area of a Triangle]
Step 2: Substitution of Given Variables
Given vertices are $(x_1, y_1) = (2, -6)$, $(x_2, y_2) = (5, 4)$, and $(x_3, y_3) = (k, 4)$. The area is given as $35$ sq units.
Substituting into the determinant form:
$35 = \pm \frac{1}{2} \begin{vmatrix} 2 & -6 & 1 \\ 5 & 4 & 1 \\ k & 4 & 1 \end{vmatrix}$
[Note: Since area is an absolute value, we must account for both positive and negative results when removing the modulus bars.]
Step 3: Evaluating the Determinant
Expand along the first row:
$\Delta = 2(4(1) - 4(1)) - (-6)(5(1) - k(1)) + 1(5(4) - k(4))$
$\Delta = 2(0) + 6(5 - k) + (20 - 4k)$
$\Delta = 30 - 6k + 20 - 4k$
$\Delta = 50 - 10k$
Step 4: Solving for $k$
From the formula $\text{Area} = \frac{1}{2} |\Delta|$:
$35 = \frac{1}{2} |50 - 10k|$
$70 = |50 - 10k|$
This yields two linear equations:
- Case 1: $50 - 10k = 70$
$-10k = 20 \implies k = -2$ - Case 2: $50 - 10k = -70$
$-10k = -120 \implies k = 12$
Step 5: Visual Representation
Final Solution: The possible values for $k$ are $12$ or $-2$.
More Questions from Class 12 Mathematics Determinants EXERCISE 4.2
- Q1(i): Find area of the triangle with vertices at the point given in each of the following: (i) $(1, 0), (6, 0), (4, 3)$
- Q1(ii): Find area of the triangle with vertices at the point given in each of the following: (ii) $(2, 7), (1, 1), (10, 8)$
- Q1(iii): Find area of the triangle with vertices at the point given in each of the following: (iii) $(–2, –3), (3, 2), (–1, –8)$
- Q2: Show that points A $(a, b + c)$, B $(b, c + a)$, C $(c, a + b)$ are collinear.
- Q3(i): Find values of $k$ if area of triangle is 4 sq. units and vertices are (i) $(k, 0), (4, 0), (0, 2)$
- Q3(ii): Find values of $k$ if area of triangle is 4 sq. units and vertices are (ii) $(–2, 0), (0, 4), (0, k)$
- Q4(i): (i) Find equation of line joining $(1, 2)$ and $(3, 6)$ using determinants.
- Q4(ii): (ii) Find equation of line joining $(3, 1)$ and $(9, 3)$ using determinants.
CBSE Solutions for Class 12 Mathematics Determinants
Chapters in CBSE - Class 12 Mathematics
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