If the tangent to the curve <math><mrow><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi></mrow></math> at the point (a, b) is also tangent to the curve y = 5x2 + 2x – 25 at the point (2, 1), then <math><mrow><mrow><mo>|</mo><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mn>9</mn><mi>b</mi></mrow><mo>|</mo></mrow></mrow></math> is equal to

y = 5x2 + 2x – 25P(2, -1)T(p) : T = 0Þ y – 1 = 10x(2) + 2(x + 2) – 50 ⇒ y = 2 2 x − 4 5 is also tangent to y = x3 – x2 + x at point (a, b)For y = x3 -x2 + x d y d x = 3 x 2 − 2 x + 1 > 0  y = 22x − 1 9 3 9 2 7  which is not tangent to the curve. 3 a 2 − 2 a + 1 = 2 2 (slope of tangent) ⇒ 3 a 2 − 2 a − 2 1 = 0 → a = 2 ± 4 + 2 5 2 6 = 2 ± 1 6 6 = 3 , − 7 3  b = 27 – 9 + 3 = 21tangent : y – 21 = 22(x – 3) ⇒ y = 22x – 45a = 3, b = 212a + 9b = 6 + 189 = 195Also, a 3 − a 2 + a = b  For a = − 7 3  b = − 3 4 3 2 7 − 4 9 9 − 7 3  = − 3 4 3 − 1 4 7 − 6 3 2 7 = − 5 5 3 2 7

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Application of Derivatives Ncert Solutions Maths class 12th Maths Class 12th

If the tangent to the curve $y = x^{3} - x^{2} + x$ at the point (a, b) is also tangent to the curve y = 5x² + 2x – 25 at the point (2, 1), then $| 2 a + 9 b |$ is equal to

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Payal Gupta | Contributor-Level 10

2 months ago

y = 5x² + 2x – 25

P(2, -1)

T_(p) : T = 0

Þ y – 1 = 10x(2) + 2(x + 2) – 50

$\Rightarrow y = 22 x - 45$ is also tangent to y = x³ – x² + x at point (a, b)

For y = x³ -x² + x

$\frac{d y}{d x} = 3 x^{2} - 2 x + 1 > 0$

y = 22x $- \frac{1939}{27}$ which is not tangent to the curve.

$3 a^{2} - 2 a + 1 = 22$ (slope of tangent)

$\Rightarrow 3 a^{2} - 2 a - 21 = 0 \to a = \frac{2 \pm \sqrt[]{4 + 252}}{6} = \frac{2 \pm 16}{6} = 3, \frac{- 7}{3}$

b = 27 – 9 + 3 = 21

tangent : y – 21 = 22(x – 3)

⇒ y = 22x – 45

a = 3, b = 21

2a + 9b = 6 + 189 = 195

Also, $a^{3} - a^{2} + a = b$

For a = $\frac{- 7}{3}$

b = $\frac{- 343}{27} - \frac{49}{9} - \frac{7}{3}$

= $\frac{- 343 - 147 - 63}{27} = \frac{- 553}{27}$

y = 5x2 + 2x – 25P(2, -1)T(p) : T = 0Þ y – 1 = 10x(2) + 2(x + 2) – 50 <math> <mrow> <mo>⇒</mo> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mn>5</mn> </mrow> </math> is also tangent to y = x3 – x2 + x at point (a, b)For y = x3 -x2 + x <math> <mrow> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>></mo> <mn>0</mn> </mrow> </math>               y = 22x <math> <mrow> <mo>−</mo> <mfrac> <mrow> <mn>1</mn> <mn>9</mn> <mn>3</mn> <mn>9</mn> </mrow> <mrow> <mn>2</mn> <mn>7</mn> </mrow> </mfrac> </mrow> </math>  which is not tangent to the curve. <math> <mrow> <mn>3</mn> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>2</mn> <mn>2</mn> </mrow> </math> (slope of tangent) <math> <mrow> <mo>⇒</mo> <mn>3</mn> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>2</mn> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>→</mo> <mi>a</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>±</mo> <mroot> <mrow> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mn>5</mn> <mn>2</mn> </mrow> <mrow></mrow> </mroot> </mrow> <mrow> <mn>6</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>±</mo> <mn>1</mn> <mn>6</mn> </mrow> <mrow> <mn>6</mn> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mfrac> <mrow> <mo>−</mo> <mn>7</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math>               b = 27 – 9 + 3 = 21tangent : y – 21 = 22(x – 3) ⇒ y = 22x – 45a = 3, b = 212a + 9b = 6 + 189 = 195Also, <math> <mrow> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mrow> </math>  For a = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>7</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math>  b = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> <mn>4</mn> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mn>7</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mn>4</mn> <mn>9</mn> </mrow> <mrow> <mn>9</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <mn>7</mn> </mrow> <mrow> <mn>3</mn> </mrow> </mfrac> </mrow> </math>  = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <mn>3</mn> <mn>4</mn> <mn>3</mn> <mo>−</mo> <mn>1</mn> <mn>4</mn> <mn>7</mn> <mo>−</mo> <mn>6</mn> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mn>7</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>5</mn> <mn>5</mn> <mn>3</mn> </mrow> <mrow> <mn>2</mn> <mn>7</mn> </mrow> </mfrac> </mrow> </math>

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If the tangent to the curve $y = x^{3} - x^{2} + x$ at the point (a, b) is also tangent to the curve y = 5x² + 2x – 25 at the point (2, 1), then $| 2 a + 9 b |$ is equal to

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If the tangent to the curve y=x3−x2+x at the point (a, b) is also tangent to the curve y = 5x2 + 2x – 25 at the point (2, 1), then |2a+9b| is equal to

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If the tangent to the curve $y = x^{3} - x^{2} + x$ at the point (a, b) is also tangent to the curve y = 5x² + 2x – 25 at the point (2, 1), then $| 2 a + 9 b |$ is equal to