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WB JEE 2021

MCQ (More than One Correct Answer)
English
Bengali
A plane meets the co-ordinate axes t the points A, B, C respectively such a way that the centroid of $$\Delta$$ABC is (1, r, r2) for some real r. If the plane passes through the point (5, 5, $$-$$12) then r =
A
$${3 \over 2}$$
B
4
C
$$-$$ 4
D
$$-$$ $${3 \over 2}$$

Explanation

Let equation of plane be

$${x \over a} + {y \over b} + {z \over c} = 1$$

Centroid $$ = \left( {{a \over 3},{b \over 3},{c \over 3}} \right)$$

$$\therefore$$ $${a \over 3} = 1,{b \over 3} = r,{c \over 3} = {r^2}$$

$$\Rightarrow$$ a = 3, b = 3r, c = 3r2

Equation of plane be

$${x \over 3} + {y \over {3r}} + {z \over {3{r^2}}} = 1$$

Now plane is passes through (5, 5, $$-$$12)

$$\therefore$$ $${5 \over 3} + {5 \over {3r}} - {{12} \over {3{r^2}}} = 1$$

$$ \Rightarrow 5{r^2} + 5r - 12 = 3{r^2}$$

$$ \Rightarrow 2{r^2} + 5r - 12 = 0$$

$$ \Rightarrow 2{r^2} + 8r - 3r - 12 = 0$$

$$ = (2r - 3)(r + 4) = 0$$

$$\Rightarrow$$ r = 3 / 2, $$-$$4
একটি তল স্থানাঙ্ক অক্ষগুলিকে যথাক্রমে এরূপ তিনটি বিন্দু A, B, C তে ছেদ করে যে $$\Delta$$ABC এর ভরকেন্দ্র হয় (1, r, r2), r এর একটি বাস্তব মানের জন্য। যদি ঐ তলটি (5, 5, $$-$$12) বিন্দুগামী হয় , তবে r হবে
A
$${3 \over 2}$$
B
4
C
$$-$$ 4
D
$$-$$ $${3 \over 2}$$

Explanation

সমতল সমীকরণ ধরা যাক

$${x \over a} + {y \over b} + {z \over c} = 1$$

ভরকেন্দ্র $$ = \left( {{a \over 3},{b \over 3},{c \over 3}} \right)$$

$$\therefore$$ $${a \over 3} = 1,{b \over 3} = r,{c \over 3} = {r^2}$$

$$\Rightarrow$$ a = 3, b = 3r, c = 3r2

সমতলের সমীকরণ হবে

$${x \over 3} + {y \over {3r}} + {z \over {3{r^2}}} = 1$$

এখন সমতল (5, 5, −12) এর মধ্য দিয়ে যায়

$$\therefore$$ $${5 \over 3} + {5 \over {3r}} - {{12} \over {3{r^2}}} = 1$$

$$ \Rightarrow 5{r^2} + 5r - 12 = 3{r^2}$$

$$ \Rightarrow 2{r^2} + 5r - 12 = 0$$

$$ \Rightarrow 2{r^2} + 8r - 3r - 12 = 0$$

$$ = (2r - 3)(r + 4) = 0$$

$$\Rightarrow$$ r = 3 / 2, $$-$$4

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