Q101 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Mathematics

If the line joining the points $A(2,3,-1)$ and $B(3,5,-3)$ is perpendicular to the line joining $C(1,2,3)$ and $D(3, y, 7)$, then $y=$

  • A
    1
  • B
    3
  • Correct answer
    C
    5
  • D
    7

Solution

$$ \begin{aligned} & \text {Given, } A=(2,3-1) \\ & \qquad B=(3,5-3) \\ & \text { DR's of } \mathbf{A B}=(3-2,5-3,-3+1)=(1,2,-2) \\ & \text { and } C=(1,2,3) \\ & \qquad D=(3, y, 7) \end{aligned} $$ $$ \therefore \mathrm{DR}^{\prime} \mathrm{s} \text { of } \mathbf{C D}=(3-1, y-2,7-3)=(2, y-2,4) $$ Given, $\mathbf{A B} \perp \mathbf{C D}$ $$ \begin{aligned} & \Rightarrow a_1 a_2+b_1 b_2+c_1 c_2=0 \\ & \Rightarrow(1)(2)+2(y-2)+(-2)(4)=0 \\ & \Rightarrow 2 y-10=0 \\ & \Rightarrow y=5 \end{aligned} $$ Hence, option (c) is correct.

Hence, the correct answer is option: 'C'.

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