If a normal to y^2=4 ax makes an angle θwith the positive direction of x -axis, perpendicular di

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If a normal to y^2=4 ax makes an angle θwith the positive...

If a normal to $\mathrm{y}^{2}=4 \mathrm{ax}$ makes an angle $\theta$ with the positive direction of $\mathrm{x}$ -axis, perpendicular distance of it from a tangent parallel to it is (a) a $\operatorname{cosec}^{2} \theta \sec \theta$ (b) a $\operatorname{cosec} \theta \sec \theta$ (c) a $\operatorname{cosec} \theta \sec ^{2} \theta$ (d) a $\operatorname{cosec} \theta \sec ^{3} \theta$

If a normal to $\mathrm{y}^{2}=4 \mathrm{ax}$ makes an angle $\theta$ with the positive direction of $\mathrm{x}$ -axis, perpendicular distance of it from a tangent parallel to it is
(a) a $\operatorname{cosec}^{2} \theta \sec \theta$
(b) a $\operatorname{cosec} \theta \sec \theta$
(c) a $\operatorname{cosec} \theta \sec ^{2} \theta$
(d) a $\operatorname{cosec} \theta \sec ^{3} \theta$

Key Concepts

Trigonometric Relationships

Trigonometric functions such as secant and cosecant relate the angle a line makes with the axes to its slope. In this context, knowing the angle ? that a normal makes with the x-axis allows the use of trigonometric identities to express geometric quantities like distances in terms of sec ? and cosec ?.

Tangent and Parallel Lines

A tangent line touches a curve at exactly one point without crossing it, characterized by the slope derived from the derivative. When a tangent is parallel to another line (like a normal in another context), they share the same slope, a concept that is key to setting up conditions for parallelism in coordinate geometry.

Perpendicular Distance Between Parallel Lines

The perpendicular distance between two parallel lines is computed using a standard formula that involves the differences in their constant terms divided by the square root of the sum of the squares of the coefficients of x and y. This concept is crucial when determining distances in problems where both lines share the same orientation.

Parabola

A parabola is a conic section with a standard equation such as y² = 4ax, which exhibits unique geometric properties including a focus, directrix, and an axis of symmetry. This equation establishes the framework for determining tangents and normals at any point on the curve.

Normal to a Curve

The normal at a point on a curve is the line perpendicular to the tangent at that point. For the parabola y² = 4ax, after obtaining the tangent's slope through differentiation, the normal's slope is found by taking the negative reciprocal, linking the algebraic form of the curve to its geometric behavior.

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