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 :
\(3+4\sqrt 2\)
\(-5+6\sqrt 2\)
\(-4+3\sqrt 2\)
\(7+6\sqrt 2\)
The Correct Option is C
Solution and Explanation
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,
\(m_2=\frac {−32±\sqrt {(32)^2+4(32)}}{64}\)
\(8m_1m_2=−4+3\sqrt 2\)
So, the correct option is (C): \(−4+3\sqrt 2\)
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Concepts Used:
Conic Sections
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
Read More: Conic Sections