Plane ax - by - cz -1 intersects axes in A - B - C respectively- If G 1-6-1-3- 1 is a centroid of - ABC- then a - b -3 c -4A- 4-3B- 4C- 2 5D- 0-6

The correct option is C 2Given plane is ax+by+cz=1 Hence, this plane intersects axes at A(1a,0,0), B(0,1b,0), C(0,0,1c) respectively. Centroid of Δ ABC is giv ...

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Standard XII

Mathematics

Equation of a Plane : Intercept Form

Plane ax + by...

Question

Plane $a x + b y + c z = 1$ intersects axes in $A, B, C$ respectively. If $G (\frac{1}{6}, - \frac{1}{3}, 1)$ is a centroid of $Δ A B C$ , then $a + b + 3 c =$ _____

\frac{4}{3}

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\frac{5}{6}

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Solution

The correct option is C $2$
Given plane is $a x + b y + c z = 1$
Hence, this plane intersects axes at $A (\frac{1}{a}, 0, 0), B (0, \frac{1}{b}, 0), C (0, 0, \frac{1}{c})$ respectively.

Centroid of $Δ A B C$ is given by
$G (\frac{1}{3 a}, \frac{1}{3 b}, \frac{1}{3 c}) \equiv (\frac{1}{6}, - \frac{1}{3}, 1)$
$\Rightarrow a = 2, b = - 1, c = \frac{1}{3}$
$∴ a + b + 3 c = 2$

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Plane ax+by+cz=1 intersects axes in A,B,C respectively. If G(16,−13,1) is a centroid of ΔABC, then a+b+3c=_____

The correct option is C 2Given plane is ax+by+cz=1 Hence, this plane intersects axes at A(1a,0,0), B(0,1b,0), C(0,0,1c) respectively. Centroid of ΔABC is given by G(13a,13b,13c)≡(16,−13,1) ⇒a=2, b=−1, c=13 ∴a+b+3c=2

Plane $a x + b y + c z = 1$ intersects axes in $A, B, C$ respectively. If $G (\frac{1}{6}, - \frac{1}{3}, 1)$ is a centroid of $Δ A B C$ , then $a + b + 3 c =$ _____