Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • If <img alt="\mathrm{\left(\frac{a}{b}\right)^{1 / 3}+\left(\frac{b}{a}\right)^{1 / 3}=\frac{\sqrt{3}}{2}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Cleft%28%5Cfrac%7

If \mathrm{\left(\frac{a}{b}\right)^{1 / 3}+\left(\frac{b}{a}\right)^{1 / 3}=\frac{\sqrt{3}}{2}}, then the angle of intersection of the parabola \mathrm{y^2=4 a x} and \mathrm{x^2=4 b y} at a point other than the origin is

Option: 1

\mathrm{\pi / 4}


Option: 2

\mathrm{\pi / 3}


Option: 3

\mathrm{\pi / 2}


Option: 4

\mathrm{\text { None of these }}


Answers (1)

best_answer

Given parabolas are \mathrm{y^2=4 a x}        \mathrm{.....(i)}  and   \mathrm{ x^2=4 b y}       \mathrm{....(ii)}

These meet at the points \mathrm{(0,0),\left(4 a^{1 / 3} b^{2 / 3}, 4 a^{2 / 3} b^{1 / 3}\right)}

Tangent to (i) at \mathrm{\left(4 a^{1 / 3} b^{2 / 3}, 4 a^{2 / 3} b^{1 / 3}\right) is \, y \cdot 4 a^{2 / 3} b^{1 / 3}=2 a\left(x+4 a^{2 / 3} b^{1 / 3}\right)}

Slope of the tangent  \mathrm{ \left(m_1\right)=\frac{2 a}{4 a^{2 / 3} b^{1 / 3}}=\frac{a^{1 / 3}}{2 b^{1 / 3}} }

Tangent to (ii) at \mathrm{\left(4 a^{1 / 3} b^{2 / 3}, 4 a^{2 / 3} b^{1 / 3}\right) is \, \, x \cdot 4 a^{1 / 3} b^{2 / 3}=2 b\left(y+4 a^{2 / 3} b^{1 / 3}\right)}

Slope of the tangent \mathrm{\left(m_2\right)=\frac{2 a^{1 / 3}}{b^{1 / 3}}}

\mathrm{\text { If } \theta \text { is the angle between the two tangents, then } \Rightarrow \tan \theta=\left|\frac{m_1-m_2}{1+m_1 m_2}\right|=\left|\frac{\frac{a^{1 / 3}}{2 b^{1 / 3}}-\frac{2 a^{1 / 3}}{b^{1 / 3}}}{1+\frac{a^{1 / 3}}{2 b^{1 / 3}} \cdot \frac{2 a^{1 / 3}}{b^{1 / 3}}}\right|}

\mathrm{ =\frac{3}{2} \cdot \frac{1}{\left(\frac{a}{b}\right)^{1 / 3}+\left(\frac{b}{a}\right)^{1 / 3}}=\frac{3}{2} \cdot \frac{1}{\frac{\sqrt{3}}{2}}=\sqrt{3} \quad ; \, \, \, \therefore \quad \theta=60^{\circ}=\frac{\pi}{3} }

Posted by

Gunjita

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Latest Question