Mathematics · Class 12 Science
Ch 14Ellipse — Class 12 Mathematics, concept-first.
The ellipse is not just a mathematical curiosity — it is, famously, the shape traced by every planet's orbit around the sun, an idea with deep roots in ancient Indian astronomical thought: verses in the Rigveda describe the sun's yearly path as an unbroken, ever-repeating course, a picture later astronomers would forma…
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Ellipse
An ellipse is the set of points whose distance from a fixed focus is a constant fraction (the eccentricity, less than 1) of its distance from a fixed directrix; in standard position this reduces to the neat equation x²/a…
Most relevant Q&A
- Find the equation of the tangent and normal to the ellipse $9x^2+16y^2=144$ at the end of the latus rectum in the first quadrant.Preview
- If P(x, y) is any point on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \ (a>b)$ whose foci are S and S' then prove that SP + S'P is a c…Preview
- Find the length of major axis, minor axis, latus rectum, eccentricity, coordinates of centre, foci and the equations of directrices of the e…Preview
- If the normal at one end of a latus rectum of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ passes through one end of the minor axis…Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
The ellipse is not just a mathematical curiosity — it is, famously, the shape traced by every planet's orbit around the sun, an idea with deep roots in ancient Indian astronomical thought: verses in t…
The Ellipse and Its Standard Equation
Start with a fixed point (the focus), a fixed line (the directrix), and a fixed positive number (the eccentricity).
Eccentricity, Foci, Directrices and the Latus Rectum
Once an ellipse is written as with , every other feature of the curve — the two foci, the two directrices, and the eccentricity itself — can be recovered directly from and , without going back to the…
The Focal-Distance Property
The most famous fact about an ellipse — the one behind the classic "pin-and-string" construction — is this:
Parametric Equations and the Eccentric Angle
The equation describes the ellipse but doesn't hand you individual points on it conveniently — solving for given involves a square root, and picking points scattered evenly around the curve by trial i…
Equation of the Tangent to the Ellipse
Take a line and an ellipse . Substituting the line into the ellipse to find their intersection points leads to a quadratic in : A line meets a conic in general at two points; it is a tangent exactly w…
Equation of the Normal to the Ellipse
The normal to the ellipse at a point is the line through perpendicular to the tangent there. From the tangent equation at , , rewriting as shows the tangent's slope is .
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Find the equation of the tangent and normal to the ellipse $9x^2+16y^2=144$ at the end of the latus rectum in the first quadrant.Preview
- Q2If P(x, y) is any point on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 \ (a>b)$ whose foci are S and S' then prove that SP + S'P is a c…Preview
- Q3Find the length of major axis, minor axis, latus rectum, eccentricity, coordinates of centre, foci and the equations of directrices of the e…Preview
- Q4If the normal at one end of a latus rectum of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ passes through one end of the minor axis…Preview
- Q5Find the eccentricity of the ellipse $3x^2+4y^2=12$.Preview
- Q6Find the equation of the tangent to the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ at the point $(a\cos\theta,\,b\sin\theta)$.Preview