If ax - by - cZ - d intersects the coordinate axis at A - B - C and the area of the triangle ABC is a 2- b 2 -2 a 2- c 2- b 2- a 2 b 2 a 2- b 2 - If a 2- c 2- b 2 are inA-P- then find the perpendicular distance of the plane from the origin-A- 1B- 2 -3-3CD- 2

The correct option is B 2√(3)/3A≡(da,0,0), B≡(0,db,0), C≡(0,0,dc) AB=d/aî d/bĵ+0k̂ BC=0î+d/bĵ d/ck̂ Area of the triangle ABC=1/2|AB×BC|=12d^2 √((a^2+b^2+c^2)) ...

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Byju's Answer

Standard XII

Mathematics

Intercept Form of a Line

If ax + by + ...

Question

If $a x + b y + c z = d$ intersects the coordinate axis at $A, B, C$ and the area of the triangle $A B C$ is $(a^{2} + b^{2}) \sqrt{\frac{2 (a^{2} + c^{2} + b^{2})}{a^{2} b^{2} (a^{2} + b^{2})}}$ . If $a^{2}, c^{2}, b^{2}$ are in A.P. then find the perpendicular distance of the plane from the origin.

\frac{2}{3}

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\frac{2 \sqrt{3}}{3}

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Solution

The correct option is C $\frac{2 \sqrt{3}}{3}$
$A \equiv (\frac{d}{a}, 0, 0), B \equiv (0, \frac{d}{b}, 0), C \equiv (0, 0, \frac{d}{c}) - - \to A B = \frac{d}{a}^i - \frac{d}{b}^j + 0^k - - \to B C = 0^i + \frac{d}{b}^j - \frac{d}{c}^k$

Area of the triangle $A B C = \frac{1}{2} | - - \to A B \times - - \to B C | = \frac{1}{2} \frac{d^{2} \sqrt{(a^{2} + b^{2} + c^{2})}}{a b c} ∴ \frac{1}{2} \frac{d^{2} \sqrt{(a^{2} + b^{2} + c^{2})}}{a b c} = (a^{2} + b^{2}) \sqrt{\frac{2 (a^{2} + c^{2} + b^{2})}{a^{2} b^{2} (a^{2} + b^{2})}} \Rightarrow \frac{d^{2}}{2 a b c} = (2 c^{2}) \frac{\sqrt{(2)}}{a b c \sqrt{(2)}} \frac{d^{2}}{c^{2}} = 4$
$\frac{d}{c} = 2$
The distance of plane from origin $= ∣ ∣ ∣ \frac{0 + 0 + 0 - d}{\sqrt{(a^{2} + b^{2} + c^{2})}} ∣ ∣ ∣ = \frac{d}{\sqrt{3} c} = \frac{2 \sqrt{3}}{3}$

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|\begin{matrix} - a (b^{2} + c^{2} - a^{2}) & 2 b^{3} & 2 c^{3} \\ 2 a^{3} & - b (c^{2} + a^{2} - b^{2}) & 2 c^{3} \\ 2 a^{3} & 2 b^{3} & - c (a^{2} + b^{2} - c^{2}) \end{matrix}| = a b c {(a^{2} + b^{2} + c^{2})}^{3}

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If ax+by+cz=d intersects the coordinate axis at A, B, C and the area of the triangle ABC is (a2+b2)√2(a2+c2+b2)a2b2(a2+b2). If a2, c2, b2 are in A.P. then find the perpendicular distance of the plane from the origin.