3D Geometry: If a variable plane, at a distance of 3 units from the origin,… | JEE Main 2017

JEE Main 2017 (Online) 9th AprilQuestion 1

If a variable plane, at a distance of $3$ units from the origin, intersects the coordinate axes at $A$, $B$ and $C$, then the locus of the centroid of $\Delta ABC$ is

3D Geometry

$\frac1{x^{2}}+\frac1{y^{2}}+\frac1{z^{2}}=1$✓ Correct answer$\frac1{x^{2}}+\frac1{y^{2}}+\frac1{z^{2}}=3$$\frac1{x^{2}}+\frac1{y^{2}}+\frac1{z^{2}}=9$$\frac1{x^{2}}+\frac1{y^{2}}+\frac1{z^{2}}=\frac19$

Solution

Plane $\frac xa+\frac yb+\frac zc=1$ at distance $3$: $\frac1{a^2}+\frac1{b^2}+\frac1{c^2}=\frac19$. Centroid $(x,y,z)=\left(\frac a3,\frac b3,\frac c3\right)$ gives $\frac1{9x^2}+\frac1{9y^2}+\frac1{9z^2}=\frac19$, so $\frac1{x^2}+\frac1{y^2}+\frac1{z^2}=1$.

3D Geometry - JEE Mathematics question (JEE Main 2017 (Online) 9th April)

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