An equation of a tangent common to the parabolas y^2=4 x and x^2=4 y is (a) x-y+1=0 (b) x+y-1=0

step 1 Hi, the question is we're given parabola's y square equals 4x and x squared equals 4 y. Find the

An equation of a tangent common to the parabolas y^2=4 x and...

Key Concepts

Parabola

A parabola is a conic section defined as the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Its standard equations, such as y² = 4ax or x² = 4ay, serve as the basis for exploring its geometric properties and the behavior of lines, curves, and other conic sections relative to it.

Tangent Line to a Parabola

A tangent line to a parabola is a straight line that touches the curve at exactly one point without crossing it. The slope of the tangent line corresponds to the derivative of the parabola at the point of contact, and its equation can often be derived using the point-slope form or by expressing the line in slope form (y = mx + c) and applying the condition for tangency.

Common Tangent

A common tangent is a single straight line that is tangent to two (or more) different curves. Finding a common tangent involves determining a line that satisfies the tangency conditions for both curves simultaneously, meaning it must touch each curve at exactly one point, typically by equating the discriminants or comparing slopes and intercepts.

Tangency Condition

The tangency condition is an algebraic criterion used to determine when a line touches a curve at exactly one point. For a curve and a line expressed in terms of their equations, this involves substituting the line’s equation into the curve’s equation, resulting in a quadratic equation. The line is tangent to the curve if and only if the discriminant of this quadratic is zero, ensuring a unique point of intersection.

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