Let for any three distinct consecutive terms | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Since $a, b, c$ are in $A.P.$,we have $2b = a + c$,which implies $a - 2b + c = 0$.This means the line $ax + by + c = 0$ always passes through the

Vedclass · Accepted Answer

$113$

Vedclass · Answer

(B) Since $a, b, c$ are in $A.P.$,we have $2b = a + c$,which implies $a - 2b + c = 0$.This means the line $ax + by + c = 0$ always passes through the fixed point $P(1, -2)$.For the system of equations to have infinitely many solutions,the determinant of the coefficient matrix $D$ must be zero.$D = \begin{vmatrix} 1 & 1 & 1 \ 2 & 5 & \alpha \ 1 & 2 & 3 \end{vmatrix} = 1(15 - 2\alpha) - 1(6 - \alpha) + 1(4 - 5) = 0$.$15 - 2\alpha - 6 + \alpha - 1 = 0 \Rightarrow 8 - \alpha = 0 \Rightarrow \alpha = 8$.Now,for infinite solutions,$D_1 = 0$ where $D_1 = \begin{vmatrix} 6 & 1 & 1 \ \beta & 5 & 8 \ 4 & 2 & 3 \end{vmatrix} = 0$.$6(15 - 16) - 1(3\beta - 32) + 1(2\beta - 20) = 0$.$-6 - 3\beta + 32 + 2\beta - 20 = 0 \Rightarrow -\beta + 6 = 0 \Rightarrow \beta = 6$.Thus,$Q = (8, 6)$.The distance $PQ = \sqrt{(8 - 1)^2 + (6 - (-2))^2} = \sqrt{7^2 + 8^2} = \sqrt{49 + 64} = \sqrt{113}$.Therefore,$(PQ)^2 = 113$.

Similar Questions

If the system of equations $x+2y+3z=3$,$4x+3y-4z=4$,and $8x+4y-\lambda z=9+\mu$ has infinitely many solutions,then the ordered pair $(\lambda, \mu)$ is equal to

Solve the system of linear equations using the matrix method:
$5x + 2y = 3$
$3x + 2y = 5$

The system of equations $x+y+z=6$,$x+2y+5z=9$,$x+5y+\lambda z=\mu$ has no solution if

The number of values of $\alpha$ for which the system of equations: $x+y+z=\alpha$,$\alpha x+2 \alpha y+3 z=-1$,and $x+3 \alpha y+5 z=4$ is inconsistent,is:

If $A$ is a matrix such that $\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] A \left[\begin{array}{ll} 1 & 1 \end{array}\right] = \left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right]$,then $A$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module