Question

8.
The distance between the points (a cos 0 + b sin 0, 0) and (0, a sin 0 - b cos 0), is
(a) a² +6² (6) a² -6² (c) a² +6² (d) a² - 62
n
a
1:1
1:
​

Answer 1

Correct Question:-

The distance between the points (a cos ∅ + b sin ∅, 0) and (0, a sin ∅ - b cos ∅), is ?

Answer:-

First point = (a cos ∅+ b sin ∅, 0)

Second point = (0, a sin ∅ - b cos ∅)

To find the distance between these two points we need to use distance formula.

The distance formula =

$\dag\sf{\sqrt{{(x_2-x_1)}^{2}+{(y_2-y_1)}^{2}}}$

Substituting the values,

$\mapsto\sf{\sqrt{{(0 - a\:cos\theta+b\:sin\theta)}^{2}+{(a\: sin\theta-b\:cos\theta - 0)}^{2}}}$

$\mapsto\sf{\sqrt{{(a\:cos\theta+b\:sin\theta)}^{2}+{(a\:sin\theta-b\:cos\theta)}^{2}}}$

$\mapsto\sf{\sqrt{{(a\:cos\theta)}^{2}+{(b\:sin\theta)}^{2}+(2\times a\:cos\theta\times b\:sin\theta) + {(a\:sin\theta)}^{2}+{(b\: cos\theta)}^{2}-(2\times a\:sin\theta\times b\:cos\theta)}}$

$\mapsto\sf{\sqrt{{(a\:cos\theta)}^{2}+{(b\:sin\theta)}^{2}+{(a\:sin\theta)}^{2}+{(b\:cos\theta)}^{2}}}$

$\mapsto\sf{\sqrt{{(a\:cos\theta)}^{2}+{(a\:sin\theta)}^{2}+{(b\:sin\theta)}^{2}+{(b\:sin\theta)}^{2}}}$

Now, removing the common,

=> a² (cos²∅+sin²∅) + b² (cos²∅+sin²∅)

We know that, sin²∅+cos²∅=1.

Hence, a² + b² is the distance between two given points.

Answer 2

Distance formula :- √(x1 - x2)^2 + (y1 - y2)^2

• x1 = (a cosθ + b sinθ)
• x2 = 0
• y1 = 0
• y2 = (a sinθ- b cosθ)

Substituing values in the formula,

$\rightarrow \: \sqrt{ {( \sf \: acos \theta + bsin \theta - 0)}^{2} + {(0 - \sf \: asin \theta - bcos \theta)}^{2} } \\ \rightarrow \: \sqrt{ {a}^{2} { \sf \: cos \theta}^{2} + {b}^{2} { \sf \: sin \theta}^{2} + \sf \: 2abcos \theta \: sin \theta + {a}^{2} { \sf \: sin \theta}^{2} + {b}^{2} { \sf \: cos\theta}^{2} - \sf \: 2absin \: \theta \: cos \: \theta } \\ \\ \bigstar \sf \: \underline{cancel \: out \: (+ 2abcos \theta \: sin \theta ) \: and \: ( - 2absin \: \theta \: cos \: \theta)} \\ \\ \rightarrow \: \sf \: \sqrt{ {a}^{2} {cos}^{2} + {a}^{2} {sin}^{2} + {b}^{2} {sin}^{2} + {b}^{2} {cos}^{2} } \\ \rightarrow \: \sf \: \sqrt{ {a}^{2} ( {cos}^{2} + {sin}^{2} ) + {b}^{2} ( {sin}^{2} + {cos}^{2} ) } \\ \\ \bigstar \sf \: \underline{ ({cos}^{2} \theta + {sin}^{2} \theta \: = 1)} \\ \\ \rightarrow \: \sqrt{ {a}^{2} (1) + {b}^{2}(1) } \\ \rightarrow \: { \sqrt{ {a}^{2} + {b}^{2} } }$