The correct answer is (A) : \(x-y=\frac{3}{2}\) Let \(y=(mx+c)\) is tangent to \(x^2=6y\) Now , \(x^2=6(mx+c)\) So, \(x^2-6mx-6c=0\) Put \(D=b^2-4ac=0\) \(⇒ c=\frac{-3}{2}m^2\) \(\therefore \) we get \(y=mx-\frac{3}{2}m^2 \)\(.....(1)\) and given hyperbola equation is \(2x^2-4y^2=9\) \(⇒\frac{x^2}{\frac{9}{2}}-\frac{y^2}{\frac{9}{4}}=1\)\(....(2)\) Since, equation (1) is a tangent of equation (2) then \(c^2=a^2m^2-b^2\) \(⇒\frac{9}{4}m^4=9m^2-\frac{9}{4}\) \(⇒m^4=2m^2-1\) \(⇒m^4-2m^2+1=0\) \(⇒(m^2-1)^2=0\) \(⇒\)\(m=±1\) Therefore , for m=1 , equation of tangent is \(x-y=\frac{3}{2}\)
When a plane intersects a cone in multiple sections, several types of curves are obtained. These curves can be a circle, an ellipse, a parabola, and a hyperbola. When a plane cuts the cone other than the vertex then the following situations may occur:
Let ‘β’ is the angle made by the plane with the vertical axis of the cone
When β = 90°, we say the section is a circle
When α < β < 90°, then the section is an ellipse
When α = β; then the section is said to as a parabola
When 0 ≤ β < α; then the section is said to as a hyperbola
