Algebra

Matrices & Determinants

Medium

core

If the area of a triangle whose vertices are $(-2, 4)$ , $(2, -6)$ and $(k, 4)$ , $(k > 0)$ is 35 squnits, then the value of k is

✅ Correct Option: 2

The triangle has vertices $(-2, 4)$ , $(2, -6)$ , and $(k, 4)$ where $k > 0$ .

The area of the triangle is 35 square units.

For a triangle with vertices $(x_1, y_1)$ , $(x_2, y_2)$ , and $(x_3, y_3)$ , the area is:

$\text{Area} = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$

Let $(x_1, y_1) = (-2, 4)$ , $(x_2, y_2) = (2, -6)$ , and $(x_3, y_3) = (k, 4)$ .

Substituting into the formula:

$35 = \frac{1}{2}|(-2)[(-6) - 4] + 2[4 - 4] + k[4 - (-6)]|$

Calculating each term:

$(-2)[(-6) - 4] = (-2)(-10) = 20$

$2[4 - 4] = 2(0) = 0$

$k[4 - (-6)] = k(10) = 10k$

The equation becomes:

$35 = \frac{1}{2}|20 + 0 + 10k|$

$35 = \frac{1}{2}|20 + 10k|$

$70 = |20 + 10k|$

The absolute value equation gives two cases:

Case 1: $20 + 10k = 70$

$10k = 50$

$k = 5$

Case 2: $20 + 10k = -70$

$10k = -90$

$k = -9$

Since $k > 0$ , the value $k = -9$ is rejected.

Therefore, $k = 5$ .

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Consider the system of equations $x+y+z=2, 2x+3y+2z=5, x+2y+\lambda z=\mu$ then which of the following statements are TRUE?

A. The system of equations has unique solution if $\lambda=2$ and $\mu\in\mathbb R$ (set of real number)

B. The system of equations is inconsistent if $\lambda=1$ and $\mu=1$

C. The system of equations is consistent if $\lambda=1$ and $\mu=1$

D. The system of equations is consistent if $\lambda=1$ and $\mu=3$

Choose the correct answer from the options given below:

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If the area of a triangle whose vertices are (−2,4)(-2, 4)(−2,4), (2,−6)(2, -6)(2,−6) and (k,4)(k, 4)(k,4), (k>0)(k > 0)(k>0) is 35 squnits, then the value of k is

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If the area of a triangle whose vertices are $(-2, 4)$ , $(2, -6)$ and $(k, 4)$ , $(k > 0)$ is 35 squnits, then the value of k is