Question:
If the area of \( \Delta ABC \) with vertices \( A(3, 1), B(-2, 1) \) and \( C(0, k) \) is \( 5 \) sq. units, then values of \( k \) are :
If the area of \( \Delta ABC \) with vertices \( A(3, 1), B(-2, 1) \) and \( C(0, k) \) is \( 5 \) sq. units, then values of \( k \) are :
Show Hint
Never forget the "absolute value" in the area formula; area problems with a given value usually result in two possible answers.
You can also use a determinant to calculate the area for higher accuracy in complex problems.
You can also use a determinant to calculate the area for higher accuracy in complex problems.
Updated On: Sep 10, 2026
- \( 3, 1 \)
- \( -1, 3 \)
- \( -1, 2 \)
- \( 0, 2 \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
The area of a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\) is given by: \[ \text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right| \] Since area is always positive, we use the absolute value.
Step 1: Substitute the given coordinates
Given: \[ (x_1,y_1)=(3,1),\quad (x_2,y_2)=(-2,1),\quad (x_3,y_3)=(0,k) \] The area of the triangle is \(5\), so: \[ 5=\frac{1}{2}\left|3(1-k)+(-2)(k-1)+0(1-1)\right| \] \[ 10=\left|3-3k-2k+2\right| \] \[ 10=|5-5k| \]
Step 2: Solve the first case
For the first case: \[ 5-5k=10 \] \[ -5k=5 \] \[ k=-1 \]
Step 3: Solve the second case
For the second case: \[ 5-5k=-10 \] \[ -5k=-15 \] \[ k=3 \]
Final Answer:
Therefore, the possible values of \(k\) are: \[ \boxed{k=-1,\ 3} \] Hence, the correct answer is Option (B).
The area of a triangle with vertices \((x_1,y_1)\), \((x_2,y_2)\), and \((x_3,y_3)\) is given by: \[ \text{Area}=\frac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right| \] Since area is always positive, we use the absolute value.
Step 1: Substitute the given coordinates
Given: \[ (x_1,y_1)=(3,1),\quad (x_2,y_2)=(-2,1),\quad (x_3,y_3)=(0,k) \] The area of the triangle is \(5\), so: \[ 5=\frac{1}{2}\left|3(1-k)+(-2)(k-1)+0(1-1)\right| \] \[ 10=\left|3-3k-2k+2\right| \] \[ 10=|5-5k| \]
Step 2: Solve the first case
For the first case: \[ 5-5k=10 \] \[ -5k=5 \] \[ k=-1 \]
Step 3: Solve the second case
For the second case: \[ 5-5k=-10 \] \[ -5k=-15 \] \[ k=3 \]
Final Answer:
Therefore, the possible values of \(k\) are: \[ \boxed{k=-1,\ 3} \] Hence, the correct answer is Option (B).
Was this answer helpful?
0
0
Top CBSE CLASS XII Mathematics Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A committee of 11 members is to be formed from 8 males and 5 females. Let m denotes the number of ways of selecting committee having atleast 6 males and n denotes the number of ways of selecting atleast 3 females then
Determine whether each of the following relations are reflexive, symmetric, and transitive.
- Relation R in the set A={1,2,3,...13,14} defined as R={(x,y): 3x-y=0}.
- Relation R in the set of N natural numbers defined as R={(x,y): y=x+5 and x<4}.
- Relation R in the set A={1,2,3,4,5,6} defined as R={(x,y): y is divisible by x}.
- Relation R in the set of Z integers defined as R={(x,y): x-y is an integer}
- Relation R in the set of human beings in a town at a particular time given by
- R={(x,y): x and y work at same place}
- R={(x,y): x and y live in the same locality}
- R={(x,y): x is exactly 7 am taller that y}
- R={(x,y): x is wife of y}
- R={(x,y):x is father of y}
Show that the relation R in the set R of real numbers, defined as
R = {(a, b): a ≤ b2 } is neither reflexive nor symmetric nor transitive.Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.
View More Questions
Top CBSE CLASS XII Area of a Triangle Questions
Top CBSE CLASS XII Questions
- 2, b, c are in A.P. and the range of determinant $\begin{vmatrix}1&1&1\\ 2&b&c\\ 4&b^{2}&c^{2}\end{vmatrix}$ is [2, 16]. Then find range of c is
- A boy of mass 50 kg is standing at one end of a, boat of length 9 m and mass 400 kg. He runs to the other, end. The distance through which the centre of mass of the boat boy system moves is
- A capillary tube of radius r is dipped inside a large vessel of water. The mass of water raised above water level is M. If the radius of capillary is doubled, the mass of water inside capillary will be
- A circular disc is rotating about its own axis. An external opposing torque 0.02 Nm is applied on the disc by which it comes rest in 5 seconds. The initial angular momentum of disc is
- A circular disc is rotating about its own axis at uniform angular velocity \(\omega.\) The disc is subjected to uniform angular retardation by which its angular velocity is decreased to \(\frac {\omega}{2}\) during 120 rotations. The number of rotations further made by it before coming to rest is
View More Questions