Let $M$ and $N$ be the number of points on the curve $y^{5}-9 x y+2 x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value of $M+N$ equals ___________.
Answer (integer)
2
Solution
<p>Here equation of curve is</p>
<p>${y^5} - 9xy + 2x = 0$ ...... (i)</p>
<p>On differentiating : $5{y^4}{{dy} \over {dx}} - 9y - 9x{{dy} \over {dx}} + 2 = 0$</p>
<p>$\therefore$ ${{dy} \over {dx}} = {{9y - 2} \over {5{y^4} - 9x}}$</p>
<p>When tangents are parallel to x-axis then $9y - 2 = 0$</p>
<p>$\therefore$ $M = 1$.</p>
<p>For tangent perpendicular to x-axis</p>
<p>$5{y^4} - 9x = 0$ ...... (ii)</p>
<p>From equation (i) and (ii) we get only one point.</p>
<p>$\therefore$ $N = 1$.</p>
<p>$\therefore$ $M + N = 2$.</p>
About this question
Subject: Mathematics · Chapter: Application of Derivatives · Topic: Tangents and Normals
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