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Let the normal at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, -3) and $(4, - 2 \sqrt[]{2})$ and given that $a - 2 \sqrt[]{2} b = 3,$ then $(a^{2} + b^{2} + a b)$ is equal to…………

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Vishal Baghel | Contributor-Level 10

a month ago

Let the equation of normal is Y – y = - $\frac{1}{m} (X - x)$

where m is slope of tangent to the given curve then

$Y - y = - \frac{d x}{d y} (X - x)$

It passes through (a, b) so b – y = $\frac{- d x}{d y} (a - x)$

=> (a – x) dx = (y – b) dy

On integration $a x - \frac{x^{2}}{2} = \frac{y^{2}}{2} - b y + c . . . . . . . . . (i)$

(ii) passes through (3, -3) & then

3a – 3b – c = 9 .(ii)

& 4a - $2 \sqrt[]{2} b$ - c = 12 .(iii)

also given $a - 2 \sqrt[]{2} b = 3 . . . . . . . . . . . . (i v)$

Solve (ii), (iii) & (iv) b = 0, a = 3

Hence a² + b² + ab = 9

Let the equation of normal is Y – y = - <math> <mrow> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <mi>m</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>  where m is slope of tangent to the given curve then  <math> <mrow> <mi>Y</mi> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>X</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>         It passes through (a, b) so b – y = <math> <mrow> <mfrac> <mrow> <mo>−</mo> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>y</mi> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> => (a – x) dx = (y – b) dyOn integration     <math> <mrow> <mi>a</mi> <mi>x</mi> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>x</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <msup> <mrow> <mi>y</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>−</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>  (ii) passes through (3, -3) &  then3a – 3b – c = 9       .(ii)& 4a - <math> <mrow> <mn>2</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> <mi>b</mi> </mrow> </math> - c = 12           .(iii)also given  <math> <mrow> <mi>a</mi> <mo>−</mo> <mn>2</mn> <mroot> <mrow> <mn>2</mn> </mrow> <mrow></mrow> </mroot> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> </math> Solve (ii), (iii) & (iv) b = 0, a = 3Hence a2 + b2 + ab = 9

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Let the normal at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, -3) and $(4, - 2 \sqrt[]{2})$ and given that $a - 2 \sqrt[]{2} b = 3,$ then $(a^{2} + b^{2} + a b)$ is equal to…………

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Let the normal at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, -3) and ( 4 , − 2 2 ) and given that a − 2 2 b = 3 , then ( a 2 + b 2 + a b ) is equal to…………

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Let the normal at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, -3) and $(4, - 2 \sqrt[]{2})$ and given that $a - 2 \sqrt[]{2} b = 3,$ then $(a^{2} + b^{2} + a b)$ is equal to…………