One of the common tangents of x^2+y^2=(a^2)/(2) and y^2=4 ax is (a) y=x+a (b) y=-x+a (c) x=y+a (

step 1 So in this question we have to find one of the common tangents for this circle and this parabola

One of the common tangents of x^2+y^2=(a^2)/(2) and y^2=4 ax...

Key Concepts

Common Tangents

Common tangents are lines that simultaneously touch two or more curves at exactly one point each. The concept of common tangents is critical when analyzing the relative positions and interactions of conic sections. Determining common tangents involves finding line equations that satisfy the tangency conditions for all of the curves involved.

Tangency Conditions Using the Discriminant

When a line is substituted into the equation of a conic section, the resulting quadratic equation in the remaining variable must have exactly one solution for the line to be tangent to the conic. This is achieved by setting the discriminant of the quadratic equation to zero. This algebraic condition is a powerful method for verifying or deriving the equation of a tangent to a conic section.

Tangent to a Circle

A tangent to a circle is a straight line that touches the circle at exactly one point. For a circle, the condition for a line to be tangent is that the perpendicular distance from the center of the circle to the line must equal the radius of the circle. This concept plays a key role in analyses involving circles since it provides a direct geometric method to determine or verify tangency.

Tangent to a Parabola

A tangent to a parabola is a line that intersects the parabola at precisely one point. For a parabola given in standard form, there exists a specific condition (or relation among the line’s parameters) that must be satisfied for tangency. Typically, substituting the line equation into the parabola’s equation yields a quadratic equation; setting its discriminant to zero ensures that the line touches the parabola at a unique point.

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