Let the lines y + 2x = <math><mrow><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mo>+</mo><mn>7</mn><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow></math> and <math><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mi>x</mi><mo>=</mo><mn>2</mn><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mo>+</mo><mn>6</mn><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow></math> be normal to a circle C : <math><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>x</mi><mo>−</mo><mi>h</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mrow><mo>(</mo><mrow><mi>y</mi><mo>−</mo><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>.</mo></mrow></math> If the line <math><mrow><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mi>y</mi><mo>−</mo><mn>3</mn><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mroot><mrow><mn>7</mn><mn>7</mn></mrow><mrow></mrow></mroot></mrow><mrow><mn>3</mn></mrow></mfrac><mo>+</mo><mn>1</mn><mn>1</mn></mrow></math> is tangent to the circle C, then the value of <math><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>5</mn><mi>h</mi><mo>−</mo><mn>8</mn><mi>k</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>5</mn><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> is equal to

y+2x=11+77 ……… (i)2y+x=211+67 ……… (ii)⇒x+y=11+1337 ……… (iii)Centre of the circle given by solving (i) & (ii)as (873, 11+573)Again 11y−3x=5773 is tangent to the circle. ∴ r = | 1 1 ( 1 1 + 5 7 3 ) − 8 7 − 5 7 7 3 1 1 + 9 | = | 1 1 − 8 7 2 0 | ∴ ( 5 h − 8 k ) 2 + 5 r 2 = 8 1 6

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Maths NCERT Exemplar Solutions Class 12th Chapter Ten Maths Class 12th Maths NCERT Exemplar Solutions Class 12th Chapter Thirteen Limits and Derivatives

Let the lines y + 2x = $\sqrt[]{11} + 7 \sqrt[]{7}$ and $2 y + x = 2 \sqrt[]{11} + 6 \sqrt[]{7}$ be normal to a circle C : ${(x - h)}^{2} + {(y - k)}^{2} = r^{2} .$ If the line $\sqrt[]{11} y - 3 x = \frac{5 \sqrt[]{77}}{3} + 11$ is tangent to the circle C, then the value of ${(5 h - 8 k)}^{2} + 5 r^{2}$ is equal to

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Payal Gupta | Contributor-Level 10

2 months ago

$y + 2 x = \sqrt[]{11} + 7 \sqrt[]{7}$ ……… (i)

$2 y + x = 2 \sqrt[]{11} + 6 \sqrt[]{7}$ ……… (ii)

$\Rightarrow x + y = \sqrt[]{11} + \frac{13}{3} \sqrt[]{7}$ ……… (iii)

Centre of the circle given by solving (i) & (ii)

$a s (\frac{8 \sqrt[]{7}}{3}, \sqrt[]{11} + \frac{5 \sqrt[]{7}}{3})$

Again $\sqrt[]{11} y - 3 x = \frac{5 \sqrt[]{77}}{3}$ is tangent to the circle.

$∴ r = | \frac{\sqrt[]{11} (\sqrt[]{11} + \frac{5 \sqrt[]{7}}{3}) - 8 \sqrt[]{7} - \frac{5 \sqrt[]{77}}{3}}{\sqrt[]{11 + 9}} | = | \frac{11 - 8 \sqrt[]{7}}{\sqrt[]{20}} |$

$∴ {(5 h - 8 k)}^{2} + 5 r^{2} = 816$

<math><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mo>+</mo><mn>7</mn><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow></math> ……… (i)<math><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mi>x</mi><mo>=</mo><mn>2</mn><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mo>+</mo><mn>6</mn><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow></math> ……… (ii)<math><mrow><mo>⇒</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mo>+</mo><mfrac><mrow><mn>1</mn><mn>3</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow></math> ……… (iii)Centre of the circle given by solving (i) & (ii)<math><mrow><mi>a</mi><mi>s</mi><mrow><mo> (</mo><mrow><mfrac><mrow><mn>8</mn><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow><mrow><mn>3</mn></mrow></mfrac><mo>, </mo><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mo>+</mo><mfrac><mrow><mn>5</mn><mroot><mrow><mn>7</mn></mrow><mrow></mrow></mroot></mrow><mrow><mn>3</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math>Again <math><mrow><mroot><mrow><mn>1</mn><mn>1</mn></mrow><mrow></mrow></mroot><mi>y</mi><mo>−</mo><mn>3</mn><mi>x</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mroot><mrow><mn>7</mn><mn>7</mn></mrow><mrow></mrow></mroot></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math> is tangent to the circle. <math> <mrow> <mo>∴</mo> <mi>r</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mrow> <mroot> <mrow> <mn>1</mn> <mn>1</mn> </mrow> <mrow></mrow> </mroot> <mrow> <mo> (</mo> <mrow> <mroot> <mrow> <mn>1</mn> <mn>1</mn> </mrow> <mrow></mrow> </mroot> <mo>+</mo> <mfrac> <mrow> <mn>5</mn> <mroot> <mrow> <mn>7</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>8</mn> <mroot> <mrow> <mn>7</mn> </mrow> <mrow></mrow> </mroot> <mo>−</mo> <mfrac> <mrow> <mn>5</mn> <mroot> <mrow> <mn>7</mn> <mn>7</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> <mrow> <mroot> <mrow> <mn>1</mn> <mn>1</mn> <mo>+</mo> <mn>9</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mn>1</mn> <mo>−</mo> <mn>8</mn> <mroot> <mrow> <mn>7</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mroot> <mrow> <mn>2</mn> <mn>0</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </mrow> </math> <math> <mrow> <mo>∴</mo> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>5</mn> <mi>h</mi> <mo>−</mo> <mn>8</mn> <mi>k</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>8</mn> <mn>1</mn> <mn>6</mn> </mrow> </math>

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Let the lines y + 2x = 11+77 and 2y+x=211+67 be normal to a circle C : (x−h)2+(y−k)2=r2. If the line 11y−3x=5773+11 is tangent to the circle C, then the value of (5h−8k)2+5r2 is equal to

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