IIT JEE Main Maths -Unit 10- Tangent and normal equations- Study Notes-New Syllabus

Ellipse: Equation of Tangent

For the standard ellipse:

\( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \)

There are three important forms of tangent: point form, parametric form, and slope form.

1. Tangent at a Point on Ellipse (Point Form)

If \( (x_1, y_1) \) lies on the ellipse, then equation of tangent is:

\( \dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1 \)

This is obtained by replacing:

• \( x^2 \to xx_1 \)
• \( y^2 \to yy_1 \)

This is the most used formula for JEE.

2. Tangent in Parametric Form

The parametric point on ellipse is:

\( (x_1, y_1) = (a\cos\theta,\ b\sin\theta) \)

Substituting in point form gives tangent:

\( \dfrac{x\cos\theta}{a} + \dfrac{y\sin\theta}{b} = 1 \)

This form is very useful in advanced JEE locus/geometry questions.

3. Tangent in Slope Form

If tangent has slope \( m \), then its equation is:

\( y = mx \pm \sqrt{a^2 m^2 + b^2} \)

Condition: All real values of slope \( m \) give a valid tangent because ellipse is a closed shape.

4. Condition of Tangency (General Line)

If line is:

\( y = mx + c \)

It is tangent to ellipse iff:

\( c^2 = a^2 m^2 + b^2 \)

This is extremely important for JEE MCQs.

5. Tangent to Vertical Ellipse

For ellipse:

\( \dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1 \)

Tangent at \( (x_1, y_1) \):

\( \dfrac{xx_1}{b^2} + \dfrac{yy_1}{a^2} = 1 \)

6. Special Tangents

• At Vertex (±a,0): equation is \( x = ±a \)
• At Co-vertex (0,±b): equation is \( y = ±b \)
• At parametric θ: \( \dfrac{x\cos\theta}{a} + \dfrac{y\sin\theta}{b} =1 \)