Equation of Tangent to a Parabola

What is Parabola in Maths?

Parabola is an equation of a specific curve, such that each point on the curve is always equidistant from a fixed point and a fixed-line. The fixed point is the parabola’s focus, and the fixed line is the directrix of the parabola. Therefore, in other words, the locus of a point that is equidistant from a given point (focus) and a given line (directrix) is called the parabola....

Parabola Shape

A parabola is a U-shaped curved line where every point on the line is at an equal distance from the focus and directrix of the parabola....

Parabola Equation

Equation of Parabola can vary depending on its orientation and the position of its vertex, but one common form is:...

Properties of Parabola

Axis of Symmetry: Line that is perpendicular to the directrix and passes through the focus is called the axis of symmetry. The parabola is symmetrical about its axis of symmetry. Vertex: Point where the parabola intersects its axis of symmetry is called the vertex. The vertex is the point where the parabola changes direction from opening upwards to opening downwards (or vice versa). Focal Length: Distance between the vertex and the focus is called the focal length. All parabolas with the same focal length are similar. Directrix: Fixed line from which any point on the parabola is the same distance as the focus. Reflective Property: If a parabola is made of a material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel (“collimated”) beam, leaving the parabola parallel to the axis of symmetry. Equation of Parabola: Equation of a parabola depends on its orientation and position. The standard equation of a parabola that opens upwards and is centered at the origin is y = x2. The standard equation of a parabola that opens downwards and is centered at the origin is y = -x2....

Standard Equation of Parabola

Standard Equation of Parabola is given as follows:...

Important Terms Related to Parabola

Focus: Point (a, 0) is called the focus of a Standard Parabola (y2 = 4ax), and it has a very special property that has various real-life applications i.e. if any light ray traveling parallel to the axis of the parabola, the parabola converge those light rays at the focus....

Derivation of Parabola Equation

Take a point P with coordinates (x, y) on the parabola which lies on the X-Y plane. By the definition of the parabola, the distance of any point on the parabola from the focus and from the directrix is equal....

Graph of Parabola

Graph of the parabola is a U-shaped curve, which can open either in an upward direction or in a downward direction. Generally, the equation of a parabola which is graphed is written in the form of y = ax2 + bx + c, where a, b, and c are constants that define the shape of the parabola....

General Equations of Parabola

General equation of a parabola is given by y = a(x – h)2 + k or x = a(y – k)2 +h where (h, k) denotes the vertex of the parabola....

Parametric Coordinates of a Parabola

For a parabola, y2 = 4ax, if we take x = at2 and y = 2at for any value of “t” they will satisfy the equation of a parabola, the coordinates (at2, 2at) is termed as parametric coordinate, and “t” is called as the parameter....

Equation of Tangent to a Parabola

Tangents are lines that touch the curve only at a single point. So a line that touches the parabola exactly at one single point is called the tangent to a parabola....

Equation of Tangent in Point Form

For the given parabola y2 = 4ax equation of the tangent at point (x1, y1) is given by:...

Equation of Tangent in Parametric Form

For the given parabola y2 = 4ax equation of the tangent at point (at2, 2at) is given by:...

Equation of Tangent in Slope Form

For the given parabola y2 = 4ax with slope m equation of the tangent at point (a/m2, 2a/m) is given by...

Pair of Tangent from an External Point

Pair of tangents from an external point to any conic is given by SS1 = T2 where for parabola y2 = 4ax, S = y2 – 4ax, S1 = y12 -4ax1 and T =  yy1 – 2a(x + x1)....

Director Circle of Parabola

Director circle is the geometric object related to the conic section and is defined as the locus of the intersection of the pair perpendicular tangent of any conic. For the parabola, the director circle is the directrix as all the perpendicular pairs of tangents of the parabola intersect each other at the directrix....

Equation of Normal to a Parabola

A line perpendicular to the tangent of the parabola at the point of tangency is known as the normal of the parabola. As this line is perpendicular to the tangent at the point of tangency to the parabola, the equation of this line can be found easily if the equation of tangent and point of tangency is given, using the concept of the equation of line perpendicular to the given line, but this is not always the case....

Equation of Normal in Slope Form

For a parabola  y2 = 4ax and m is the slope of normal at the point of contact (am2, -2am), the equation of normal is given by:...

Equation of Normal in Point Form

For a parabola  y2 = 4ax, equation of normal at (x1, y1) is given as follows:...

Equation of Normal in Parametric Form

For a parabola y2 = 4ax, the equation of normal at the point (at2, 2at) [where t is the parameter] is given as follows:...

Parabola Formulas

Some important parabola formulas are added in the table below:...

Parabola Solved Examples

Example 1: Find coordinates of the focus, axis, the equation of the directrix, and latus rectum of the parabola y2 = 16x....

Practice Questions on Parabola

Q1. Find the vertex, focus, and directrix of the parabola with the equation y = x2 – 4x + 3y = x2 – 4x + 3....

Parabola in Maths – FAQs

What is Parabola?...