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Yukti Vault

3D Geometry - JEE Mathematics question (JEE Advanced 2021 Paper 1)

Comprehension · Questions 1–1

Read the passage given below and answer the questions that follow.

Let $\alpha,\beta$ and $\gamma$ be real numbers such that the system of linear equations $$x+2y+3z=\alpha$$ $$4x+5y+6z=\beta$$ $$7x+8y+9z=\gamma-1$$ is consistent. Let $|M|$ represent the determinant of the matrix $$M=\begin{bmatrix}\alpha&2&\gamma\\ \beta&1&0\\ -1&0&1\end{bmatrix}$$ Let $P$ be the plane containing all those $(\alpha,\beta,\gamma)$ for which the above system of linear equations is consistent, and $D$ be the square of the distance of the point $(0,1,0)$ from the plane $P$.
JEE Advanced 2021 Paper 1Question 1
The value of $D$ is ___ .
3D Geometry

Answer$1.5$

Solution

Consistency needs $R_1-2R_2+R_3$ to vanish on the left: $\alpha-2\beta+\gamma-1=0$. This is plane $P$. Distance of $(0,1,0)$ is $|-2-1|/\sqrt6$, so $D=9/6=1.5$.

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