A and B are points of the parabola y=x^2. The tangents at A...

$A$ and $B$ are points of the parabola $y=x^{2}$. The tangents at $A$ and $B$ meet at $C$. The mediam of the triangle $A B C$ from $C$ has length ' $m^{\prime}$ units. Find the area of the triangle in terms of ' $m$ '. Solution: $y=x^{2}$ $\frac{d y}{d x}=2 x$ tangent $\mathrm{BC} \Rightarrow y-x_{1}^{2}=2 x_{1}\left(x-x_{1}\right)$ $y=2 x x_{1}-x_{1}^{2}$ similarly tangent $A C$ $y=2 x x_{2}-x_{2}^{2}$ point of intersection (1) and (2) is $C$ $2 x x_{1}-x_{1}^{2}=2 x x_{2}-x_{2}^{2}$Now $\because \mathrm{CM}=m$ $\frac{x_{1}+x_{2}}{2}-x_{1} x^{2}=m$ $\frac{\left(x_{1}-x_{2}\right)^{2}}{2}=m$ $\left(x_{1}-x_{2}\right)^{2}-2 m$ $=\frac{1}{4}(2 m)^{3 / 2}=\frac{m^{3 / 2}}{\sqrt{3}}$ Ans

$A$ and $B$ are points of the parabola $y=x^{2}$. The tangents at $A$ and $B$ meet at $C$. The mediam of the triangle $A B C$ from $C$ has length ' $m^{\prime}$ units. Find the area of the triangle in terms of ' $m$ '.
Solution: $y=x^{2}$ $\frac{d y}{d x}=2 x$
tangent $\mathrm{BC} \Rightarrow y-x_{1}^{2}=2 x_{1}\left(x-x_{1}\right)$
$y=2 x x_{1}-x_{1}^{2}$
similarly tangent $A C$ $y=2 x x_{2}-x_{2}^{2}$
point of intersection (1) and (2) is $C$ $2 x x_{1}-x_{1}^{2}=2 x x_{2}-x_{2}^{2}$Now $\because \mathrm{CM}=m$
$\frac{x_{1}+x_{2}}{2}-x_{1} x^{2}=m$
$\frac{\left(x_{1}-x_{2}\right)^{2}}{2}=m$
$\left(x_{1}-x_{2}\right)^{2}-2 m$
$=\frac{1}{4}(2 m)^{3 / 2}=\frac{m^{3 / 2}}{\sqrt{3}}$ Ans

Key Concepts

Median of a Triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Medians are important in triangle geometry as they are used to analyze balance points and relationships between different elements of a triangle.

Intersection of Tangents

The point where two tangent lines to a single curve intersect is determined by solving their equations simultaneously. This process involves equating the tangent line equations, a technique widely used in analytic geometry to find common points between linear equations.

Area of a Triangle

The area of a triangle can be calculated using various formulas, including those from basic geometry and coordinate methods. Relating specific lengths, such as medians or distances derived from geometric constructions, to the overall area is a common application in analytic and Euclidean geometry.

Parabola

A parabola is a conic section represented by a quadratic equation. Its geometric properties and symmetry make it a fundamental subject in coordinate geometry and calculus, where points on the curve satisfy a specific quadratic relationship.

Tangent to a Curve

The tangent to a curve at a point is a straight line that touches the curve at that point and has the same instantaneous slope as the curve. This concept is derived from calculus using the derivative, which provides the slope of the curve at any given point.

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