Question

Assertion :The equation of the plane through the intersection of the planes $x-2y+3z=6$ & $4x+3y-4z-2=0$ and containing the point $(1,1,1)$ is $17x+10y-13z=14$ Reason: The equation of the plane through the line of intersection of the planes $P_1=0$ & $P_2=0$ is $P_1+\lambda P_2=0,\lambda \ne 0$

• A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion
• B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion
• C. Assertion is true but Reason is false
• D. Assertion is false but Reason is true

Answer 1

The correct option is A Both Assertion & Reason are individually true & Reason is correct explanation of Assertion

Direction of line of intersection $\overrightarrow{n_1}\times \overrightarrow{n_2}$ where $\overrightarrow{n_1}$ and $\overrightarrow{n_2}$ are directions of normal to plane

$\left|\begin{array}{ccc}i& j& k\\ 1& -2& 3\\ 4& 3& -4\end{array}\right|$

$=(8-9)i-j(-4-12)+k(3+8)$

$=-\hat{i}+16\hat{j}+11\hat{k}$

The equation of plane through the lines of intersection of planes $P_{1}and{P}_{2}is{P}_1+\lambda P_2$

$x-2y+3z-6+\lambda (4x+3y-4z-2)=0$

$x(1+4\lambda )+y(3\lambda -2)+z(3-4\lambda )-6-2\lambda =0$

$(1,1,1)$ satisfies the above equation

$(1+4\lambda )+(3\lambda -2)+(3-4\lambda )-6-2\lambda =0$

$\lambda =4$

equation of plane is $17x+10y-13z=14$