First slide

Direction ratios and direction cosines

Question

The direction cosines x,y,z of a line are connected by a relation ${(x + y)}^{2} - a^{2} = 2 x y - z^{2}$ Then $\sqrt{a^{2} + 5 a + 2019} =$

Moderate

A

1
B

19
C

53
D

45

Solution

If $〈x, y, z〉$ are direction cosines of a line then $x^{2} + y^{2} + z^{2} = 1$
Consider the equation ${(x + y)}^{2} - a^{2} = 2 x y - z^{2}$
Expand and then simplify
$\begin{matrix} x^{2} + y^{2} + 2 x y - a^{2} = 2 x y - z^{2} \Rightarrow x^{2} + y^{2} + z^{2} = a^{2} \Rightarrow a^{2} = 1 \end{matrix}$
It implies that $a = 1$
Hence the value of $\sqrt{a^{2} + 5 a + 2019}$ is $\sqrt{1 + 5 + 2019} = \sqrt{2025} = 45$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App