Question

Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 =512.
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Answer 1

If the curves $4x=y^2$ and $4xy=k$ cut at right angles, it means the tangents of these two curves drawn at the point of intersection should be perpendicular to each other.

Since the tangents are perpendicular, the slope of one among them should be the negative reciprocal of the other one.

Let us consider the first curve.

$\longrightarrow 4x=y^2$

Finding slope by differentiating the equation wrt x,

$\longrightarrow 4=2y\,y'$

$\longrightarrow y'=\dfrac{2}{y}$

Taking $y'=m_1,$

$\longrightarrow m_1=\dfrac{2}{y}$

Let us consider the second curve.

$\longrightarrow 4xy=k$

Finding slope by differentiating the equation wrt x,

$\longrightarrow 4(y+xy')=0$

$\longrightarrow y+xy'=0$

$\longrightarrow y'=-\dfrac{y}{x}$

Taking $y'=m_2,$

$\longrightarrow m_2=-\dfrac{y}{x}$

As we said earlier, the slope of one among them should be the negative reciprocal of the other.

$\longrightarrow m_1=-\dfrac{1}{m_2}$

$\longrightarrow\dfrac{2}{y}=\dfrac{x}{y}$

$\longrightarrow x=2$

Putting value of $x$ in equation of first curve,

$\longrightarrow 4\times2=y^2$

$\longrightarrow y^2=8$

Squaring equation of second curve,

$\longrightarrow (4xy)^2=k^2$

$\longrightarrow k^2=16x^2y^2$

Putting values of $x$ and $y^2,$

$\longrightarrow k^2=16\times2^2\times8$

$\longrightarrow\underline{\underline{k^2=512}}$

Hence Proved!