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  4. The curve x2 - xy + y2 =27 has tangents parallel to x-axis at

Q. The curve $x^2 - xy + y^2 =27$ has tangents parallel to $x-axis$ at

COMEDKCOMEDK 2014Application of Derivatives Report Error

Solution:

Given equation of curve is
$x^2 - xy + y^2 = 27 $ ... (i)
Taking derivative w.r.t. $'x'$ ori both sides
$\Rightarrow 2x -x \frac{dy}{dx} - y + 2y \frac{dy}{dx} = 0 $
$\Rightarrow \:\:\: \frac{dy}{dx} \left(2y -x\right)=y -2x $
$\Rightarrow \frac{dy}{dx} = \frac{y-2x}{2y-x} $
Since, curve has tangent parallel to x-axis
$\therefore $ slope of tangent= 0
$\Rightarrow \frac{dy}{dx} = 0 \Rightarrow \:\:\:\: \frac{y-2x}{2y-x} = 0 $
$\Rightarrow y -2x$ .....(ii)
\ Now solving (i) and (ii) we get,
$ x^{2} -2x^{2}+4x^{2} =27 $
$\Rightarrow 3x^{2}=27 \Rightarrow x = \pm3$
For $x = 3, y = 6$ and $x = -3, y = -6$
$\therefore $ Points are (3, 6) and (-3, -6)

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1. Let $ 𝑓 \left(x\right) = 𝑥 + log_{e} x − 𝑥 log_{e} 𝑥, 𝑥 ∈ \left(0,∞\right).$
• Column 1 contains information about zeros of $𝑓 (𝑥) , 𝑓'(𝑥)$ and $f ''(𝑥)$.
• Column 2 contains information about the limiting behavior of $𝑓(𝑥) , 𝑓 ' (𝑥)$ and $f ''(𝑥)$ at infinity.
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