(D) The given equations of the planes are:$x - cy - bz = 0$$-cx + y - az = 0$$-bx - ay + z = 0$For these planes to pass through a single line,the dete

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(D) The given equations of the planes are:$x - cy - bz = 0$$-cx + y - az = 0$$-bx - ay + z = 0$For these planes to pass through a single line,the determinant of the coefficients must be zero:$\begin{vmatrix} 1 & -c & -b \\ -c & 1 & -a \\ -b & -a & 1 \end{vmatrix} = 0$Expanding the determinant along the first row:$1(1 - a^2) - (-c)(-c - ab) + (-b)(ac + b) = 0$$1 - a^2 - c^2 - abc - abc - b^2 = 0$$1 - a^2 - b^2 - c^2 - 2abc = 0$$a^2 + b^2 + c^2 + 2abc = 1$

Class 12 Mathematics 3 and 4 .Determinants and Matrices Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

Medium

The planes $x = cy + bz, y = az + cx, z = bx + ay$ pass through one line,if

A
$a + b + c = 0$
B
$a + b + c = 1$
C
$a^2 + b^2 + c^2 = 1$
D
$a^2 + b^2 + c^2 + 2abc = 1$

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3 and 4 .Determinants and Matrices

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Expansion of determinants, Solution of equation in the form of determinants and area of triangle and Equation of Line

The value of $x,$ if $\left| \begin{matrix} -x & 1 & 0 \\ 1 & -x & 1 \\ 0 & 1 & -x \end{matrix} \right| = 0$ is equal to

Easy

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Evaluate $\left|\begin{array}{ccc}1 & x & y \\ 1 & x+y & y \\ 1 & x & x+y\end{array}\right|$

Medium

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If $a_{1}, a_{2}, a_{3}, \ldots, a_{9}$ are in $AP$,then the value of $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is

MediumKCET 2020

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If $\left| \begin{array}{ccc} 3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8 \end{array} \right| = 0$,then the values of $x$ are

Difficult

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If $A = \begin{bmatrix} 1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1 \end{bmatrix}$,then for all $\theta \in \left( \frac{3\pi}{4}, \frac{5\pi}{4} \right)$,$\det(A)$ lies in the interval:

DifficultJEE MAIN 2019

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The planes $x = cy + bz, y = az + cx, z = bx + ay$ pass through one line,if

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The value of $x,$ if $\left| \begin{matrix} -x & 1 & 0 \\ 1 & -x & 1 \\ 0 & 1 & -x \end{matrix} \right| = 0$ is equal to

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