JEE MAIN - Mathematics (2021 - 17th March Evening Shift - No. 7)
Let O be the origin. Let $$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k$$ and $$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$$, x, y$$\in$$R, x > 0, be such that $$\left| {\overrightarrow {PQ} } \right| = \sqrt {20} $$ and the vector $$\overrightarrow {OP} $$ is perpendicular $$\overrightarrow {OQ} $$. If $$\overrightarrow {OR} $$ = $$3\widehat i + z\widehat j - 7\widehat k$$, z$$\in$$R, is coplanar with $$\overrightarrow {OP} $$ and $$\overrightarrow {OQ} $$, then the value of x2 + y2 + z2 is equal to :
2
9
7
1
Explanation
$$\overrightarrow {OP} = x\widehat i + y\widehat j - \widehat k\,$$
$$\overrightarrow {OP} \bot \overrightarrow {OQ} $$
$$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$$
$$\overrightarrow {PQ} = \left( { - 1 - x} \right)\widehat i + \left( {2 - y} \right)\widehat j + \left( {3x + 1} \right)\widehat k$$
$$\left| {\overrightarrow {PQ} } \right| = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}} $$
$$\sqrt {20} = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}} $$
20 = 1 + x2 + 2x + 4 + y2 $$-$$ 4y + 9x2 + 1 + 6x
20 = 10x2 + y2 + 8x + 6 $$-$$ 4y
20 = 10x2 + 4x2 + 8x + 6 $$-$$ 8x
14 = 14x2 $$ \Rightarrow $$ x2 = 1
Also, $$\overrightarrow {OP} .\,\overrightarrow {OQ} = 0$$
$$ - x + 2y - 3x = 0$$
$$4x = 2y$$
y = 2x
$$ \therefore $$ y2 = 4x2 $$ \Rightarrow $$ y2 = 4
x = 1 as x > 0 and y = 2
$$ \therefore $$ $$\left| {\matrix{ x & y & { - 1} \cr { - 1} & 2 & {3x} \cr 3 & z & { - 7} \cr } } \right| = 0$$
$$ \Rightarrow $$ $$\left| {\matrix{ 1 & 2 & { - 1} \cr { - 1} & 2 & 3 \cr 3 & z & { - 7} \cr } } \right|$$ = 0
$$ \Rightarrow $$ 1($$-$$14 $$-$$3z) $$-$$ 2(7 $$-$$ 9) $$-$$ 1($$-$$z $$-$$6) = 0
$$ \Rightarrow $$ $$-$$14 $$-$$3z + 4 + z + 6 = 0
$$ \Rightarrow $$ 2z = $$-$$4 $$ \Rightarrow $$ z = $$-$$2
$$ \therefore $$ x2 + y2 + z2 = 9
$$\overrightarrow {OP} \bot \overrightarrow {OQ} $$
$$\overrightarrow {OQ} = - \widehat i + 2\widehat j + 3x\widehat k$$
$$\overrightarrow {PQ} = \left( { - 1 - x} \right)\widehat i + \left( {2 - y} \right)\widehat j + \left( {3x + 1} \right)\widehat k$$
$$\left| {\overrightarrow {PQ} } \right| = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}} $$
$$\sqrt {20} = \sqrt {{{\left( { - 1 - x} \right)}^2} + {{\left( {2 - y} \right)}^2} + {{\left( {3x + 1} \right)}^2}} $$
20 = 1 + x2 + 2x + 4 + y2 $$-$$ 4y + 9x2 + 1 + 6x
20 = 10x2 + y2 + 8x + 6 $$-$$ 4y
20 = 10x2 + 4x2 + 8x + 6 $$-$$ 8x
14 = 14x2 $$ \Rightarrow $$ x2 = 1
Also, $$\overrightarrow {OP} .\,\overrightarrow {OQ} = 0$$
$$ - x + 2y - 3x = 0$$
$$4x = 2y$$
y = 2x
$$ \therefore $$ y2 = 4x2 $$ \Rightarrow $$ y2 = 4
x = 1 as x > 0 and y = 2
$$ \therefore $$ $$\left| {\matrix{ x & y & { - 1} \cr { - 1} & 2 & {3x} \cr 3 & z & { - 7} \cr } } \right| = 0$$
$$ \Rightarrow $$ $$\left| {\matrix{ 1 & 2 & { - 1} \cr { - 1} & 2 & 3 \cr 3 & z & { - 7} \cr } } \right|$$ = 0
$$ \Rightarrow $$ 1($$-$$14 $$-$$3z) $$-$$ 2(7 $$-$$ 9) $$-$$ 1($$-$$z $$-$$6) = 0
$$ \Rightarrow $$ $$-$$14 $$-$$3z + 4 + z + 6 = 0
$$ \Rightarrow $$ 2z = $$-$$4 $$ \Rightarrow $$ z = $$-$$2
$$ \therefore $$ x2 + y2 + z2 = 9
Comments (0)
