Question:
A triangle is formed by the tangents at the point \((2,2)\) on the curves \(π¦^2=2π₯\) and \(π₯^2+π¦^2=4π₯\), and the line \(π₯+π¦+2=0\). If \(π\) is the radius of its circumcircle, then \(π^2\) is equal to ____ .
A triangle is formed by the tangents at the point \((2,2)\) on the curves \(π¦^2=2π₯\) and \(π₯^2+π¦^2=4π₯\), and the line \(π₯+π¦+2=0\). If \(π\) is the radius of its circumcircle, then \(π^2\) is equal to ____ .
Updated On: Mar 12, 2026
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Correct Answer: 10
Solution and Explanation
(A)For \( y^2 = 2x \), the slope of the tangent at \( (2, 2) \) is:
\[
\frac{dy}{dx} = \frac{1}{2y} = \frac{1}{4}.
\]
Equation of tangent:
\[
y - 2 = \frac{1}{4}(x - 2).
\]
(B)For \( x^2 + y^2 = 4x \), the slope of the tangent at \( (2, 2) \) is:
\[
\frac{dy}{dx} = -\frac{x - 2}{y}.
\]
Equation of tangent:
\[
y - 2 = -\frac{1}{2}(x - 2).
\]
(C)The circumcircle passes through the vertices of the triangle formed by the two tangents and the given line \( x + y + 2 = 0 \). Using the circumcircle formula and coordinates, solve for \( r^2 = 10 \).
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Concepts Used:
Coordinate Geometry
Coordinate geometry, also known as analytical geometry or Cartesian geometry, is a branch of mathematics that combines algebraic techniques with the principles of geometry. It provides a way to represent geometric figures and solve problems using algebraic equations and coordinate systems.
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher RenΓ© Descartes.
The central idea in coordinate geometry is to assign numerical coordinates to points in a plane or space, which allows us to describe their positions and relationships using algebraic equations. The most common coordinate system is the Cartesian coordinate system, named after the French mathematician and philosopher RenΓ© Descartes.