Exams Joint Entrance Examination Mathematics Matrices and Determinants Question 472 Question 472 of 472 Medium MCQ
Let the system of equations :
\[ 2 x + 3 y + 5 z = 9 7 x + 3 y − 2 z = 8 12 x + 3 y − ( 4 + λ ) z = 16 − μ \begin{aligned} & 2 x+3 y+5 z=9 \\ & 7 x+3 y-2 z=8 \\ & 12 x+3 y-(4+\lambda) z=16-\mu \end{aligned} 2 x + 3 y + 5 z = 9 7 x + 3 y − 2 z = 8 12 x + 3 y − ( 4 + λ ) z = 16 − μ \]
have infinitely many solutions. Then the radius of the circle centred at ( λ , μ ) (\lambda, \mu) ( λ , μ ) and touching the line 4 x = 3 y 4 x=3 y 4 x = 3 y is :
Solutions · 1 Post your solution Official solution
\[ \begin{aligned}
& \Delta=\left|\[\begin{array}{|ccccccc|}2 & 3 & 5 \\ 7 & 3 & -2 \\ 12 & 3 & -(4+\lambda)\end{array}\end{aligned} \]\right| \\
\& =2(-12-3 \lambda+6)-3(-28-7 \lambda+24)+5(21-36) \\
\& =-12-6 \lambda+12+21 \lambda-75 \\
\& =15 \lambda-75 \\
\& \Rightarrow 15 \lambda-75=0 \\
\& \Rightarrow \lambda=5 \\
\& \Delta\_1=\left|\[ 9 3 5 8 3 − 2 16 − μ 3 − 9 \begin{array}{|ccccccc|}9 & 3 & 5 \\ 8 & 3 & -2 \\ 16-\mu & 3 & -9\end{array} 9 8 16 − μ 3 3 3 5 − 2 − 9 \]\right| \\
\& =9(-27+6)-3(-72+32-2 \mu)+5(24-48+3 \mu) \\
\& =-189+120+6 \mu-120+15 \mu \\
\& =21 \mu-189=0 \\
\& \Rightarrow \mu=9
\]
\[ ∴ r = ∣ 4 ( 5 ) − 3 ( 9 ) ( 4 ) 2 + ( 3 ) 2 ∣ r = 7 5 \begin{aligned}
& \therefore r=\left|\frac{4(5)-3(9)}{\sqrt{(4)^2+(3)^2}}\right| \\
& r=\frac{7}{5}
\end{aligned} ∴ r = ( 4 ) 2 + ( 3 ) 2 4 ( 5 ) − 3 ( 9 ) r = 5 7 \] Answer the question to unlock