The line passes through the centre of circle x2 + y2 – 16x – 4y = 0, it interacts with the positive coordinate axis at A & B. Then find the minimum value of OA + OB, where O is origin.

(y – 2) = m (x – 8)⇒ x-intercept⇒ <!- [if gte mso 9]><xml> </xml><! [endif]-> ( − 2 m + 8 ) ⇒ y-intercept⇒ (–8m + 2)⇒ OA + OB = − 2 m 2 <!- [if gte mso 9]><xml> </xml><! [endif]-> + 8 – 8m + 2 f ' ( m ) = 2 m 2 − 8 = 0 <!- [if gte mso 9]><xml> </xml><! [endif]->-> m 2 = 1 4 -> m = − 1 2 -> f ( − 1 2 ) = 1 8 ->Minimum = 18

The line passes through the centre of circle x2 + y2 – 16x – 4y = 0, it interacts with the positive coordinate axis at A &amp; B. Then find the minimum value of OA + OB, where O is origin.

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Maths NCERT Exemplar Solutions Class 11th Chapter Eight Straight Lines Ncert Solutions Maths class 11th Maths Class 11th

The line passes through the centre of circle x² + y² – 16x – 4y = 0, it interacts with the positive coordinate axis at A & B. Then find the minimum value of OA + OB, where O is origin.

Option 1 -

Option 2 -

Option 3 -

Option 4 -

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alok kumar singh | Contributor-Level 10

a month ago

Correct Option - 1

Detailed Solution:

(y – 2) = m (x – 8)

⇒ x-intercept

⇒ $(\frac{- 2}{m} + 8)$

⇒ y-intercept

⇒ (–8m + 2)

⇒ OA + OB = $\frac{- 2}{m^{2}}$ + 8 – 8m + 2

$f' (m) = \frac{2}{m^{2}} - 8 = 0$

-> $m^{2} = \frac{1}{4}$

-> $m = \frac{- 1}{2}$

-> $f (\frac{- 1}{2}) = 18$

->Minimum = 18

(y – 2) = m (x – 8)⇒   x-intercept⇒   <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1820304504"> </o:OLEObject></xml><! [endif]->  <math> <mrow> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mi>m</mi> </mrow> </mfrac> <mo>+</mo> <mn>8</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math> ⇒   y-intercept⇒   (–8m + 2)⇒   OA + OB = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1820304505"> </o:OLEObject></xml><! [endif]-> + 8 – 8m + 2 <math> <mrow> <mi>f</mi> <mo>'</mo> <mrow> <mo> (</mo> <mrow> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mo>−</mo> <mn>8</mn> <mo>=</mo> <mn>0</mn> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1820304506"> </o:OLEObject></xml><! [endif]->-> <math> <mrow> <msup> <mrow> <mi>m</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math> -> <math> <mrow> <mi>m</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math> -> <math> <mrow> <mi>f</mi> <mrow> <mo> (</mo> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mn>8</mn> </mrow> </math> ->Minimum = 18

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Kindly consider the following figure

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$h = \frac{c o s θ + 3}{2}$

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The line passes through the centre of circle x² + y² – 16x – 4y = 0, it interacts with the positive coordinate axis at A & B. Then find the minimum value of OA + OB, where O is origin.

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