If \(y = m_1x + c_1 \)and \(y = m_2x + c_2\), \(m_1≠m_2\) are two common tangents of circle \(x_2 + y_2 = 2\) and parabola \(y^2 = x\), then the value of \(8|m_1m_2|\) is equal to :
Suppose, tangent to \(y^2 = x\) be \(y=mx+\frac {1}{4m}\) For tangent to circle, \(|\frac {\frac 14m}{\sqrt {1+m^2}}|=\sqrt 2\) \(32m^4 + 32m^2 – 1 = 0\) According to the Sridharacharya formula, \(m_2=\frac {−32±\sqrt {(32)^2+4(32)}}{64}\) \(8m_1m_2=−4+3\sqrt 2\)
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
