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Perpendicular distance of a point from a plane - class-XII
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The length of the perpendicular drawn from the points $(5,4,-1)$ to the line $\overline r = \widehat i + \lambda \left( {2\widehat i + 9\widehat i + 5\widehat k} \right)$ is
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A
$\sqrt { \frac { { 2109 } }{ { 110 } } } $
💡 Explanation:
According to the question:
$\begin{array}{l} let, the, point, (5,4,-1), , be, , P, and, the, point, through, which, the, , line, passes, , be, , Q, (1,0,0)., , , , \ the, line, is, parallel, to, the, vector:, , \overrightarrow { r } =\left( { 2\hat { i } +9\hat { i } +5\hat { k } } \right) \ Now, \ \overrightarrow { PQ } =-4\hat { i } -4\widehat { j } +\hat { k } \ \therefore , , , \overrightarrow { r, } , \times \overrightarrow { PQ } =\left| \begin{array}{l} , , \hat { i } , , , , , , , , , , , , , , \widehat { j } , , , , , , , , , , , \widehat { k } \ , , 2, , , , , , , , , , , , , , 9, , , , , , , , , , , 5, , \ , -4, , , , , , , -4, , , , , , , , , , 1 \end{array} \right| , , , , , , , , \ , , , , , , , , , , , , , , , , , , , , , =, 29, \hat { i } , -, , 22, \widehat { j } , +28, \widehat { k } \ \Rightarrow \left| { , \overrightarrow { r, } , \times \overrightarrow { PQ } } \right| =\sqrt { , { { (29) }^{ 2 } }, +{ { (-, , 22) }^{ 2 } }, +{ { (28) }^{ 2 } } } \ , , , , , , , , , , , , , , , , , , , , , , , , , =\sqrt { 841+484+784 } =\sqrt { 2109 } \ \left| { \overrightarrow { r, } , } \right| =\sqrt { { 2^{ 2 } }+{ 9^{ 2 } }+{ 5^{ 2 } } } \ , , , , , , , =\sqrt { 4+81+25 } =\sqrt { 110 } \ d=\frac { { , \left| { \overrightarrow { r, } , \times \overrightarrow { PQ } } \right| } }{ { \left| { \overrightarrow { r, } , } \right| } } =\frac { { \sqrt { 2109 } } }{ { \sqrt { 110 } } } =\sqrt { \frac { { 2109 } }{ { 110 } } } \ so, that, the, correct, option, is, C. \end{array}$