Let a smooth curve y = f(x) be such that the slope of the tangent at any point (x, y) on it is directly proportional to $\left(\frac{-y}{x}\right)$ . If the curve passes through the points (1, 2) and (8, 1), then $\left|y\left(\frac{1}{8}\right)\right|$ is equal to

<p>Slope of any point P (x, y) to y = f (x) is&nbsp;<math><mrow><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mi>k</mi><mfrac><mrow><mi>y</mi></mrow><mrow><mi>x</mi></mrow></mfrac></mrow></math></p><p><math><mrow><mo>⇒</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>y</mi></mrow></mfrac><mo>+</mo><mi>k</mi><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></math></p><p>Solving the equation the curve is xky = c</p><p>It passes (1, 2) c = 2 xky = 2 again it passes (8, 1) 8k = 2 k =&nbsp;<math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mrow></math></p><p><math><mrow><mo>∴</mo></mrow></math>&nbsp;the equation of curve is&nbsp;<math><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></math>&nbsp;y = 2 …… (i)</p><p><math><mrow><mo>∴</mo><mrow><mo>|</mo><mrow><mi>y</mi><mrow><mo> (</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mo>|</mo></mrow><mo>=</mo><mrow><mo>|</mo><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><msup><mrow><mrow><mo> (</mo><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>8</mn></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup></mrow></mfrac></mrow><mo>|</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></p>

0 Upvotes 0 Upvotes 0 Downvotes

• • Report Abuse
• Comment