Question

If the pair of equations X+Y=root2 and X sin theta +Y cos theta = 1 has a infinitely many solutions then find the value of theta

Answer 1

this is required answer
q

Answer 2

Given:

The pair of equations X + Y = √2 and X sin theta + Y cos theta = 1 has infinitely many solutions.

To find:

The value of theta.

Solution:

To solve this question, we should know that if the pair of equations ax + by = c and mx + ny = k has infinitely many solutions, then

$\frac{a}{m} = \frac{b}{n} = \frac{c}{k}$

Thus, as given, we have a pair of equations

X + Y = √2 and X sin theta + Y cos theta = 1 which

infinitely many solutions, so we have,

$\frac{1}{sin \:θ } = \frac{1}{cos \: θ} = \frac{ \sqrt{2} }{1}$

$\frac{1}{sin \:θ } = \frac{ \sqrt{2} }{1} \: and \: \frac{1}{cos \: θ} = \frac{ \sqrt{2} }{1}$

$sin \:θ = \frac{1}{ \sqrt{2} } \: and \: cos \:θ = \frac{1}{\sqrt{2} } \: (i)$

From trigonometric ratios, we know that

$sin 45° = cos 45° = \frac{1}{ \sqrt{2} } \: (ii)$

So, comparing (i) and (ii), we have,

$θ = 45°$

Hence, the value of theta (θ) is 45°.