Where Mindful Learning Begins
Where Mindful Learning Begins
Free, step-by-step NCERT Solutions for all four exercises of this chapter — the circle, parabola, ellipse and hyperbola: their standard equations, foci, vertices, axes, eccentricity and latus rectum — plus the Miscellaneous Exercise where these curves turn up in reflectors, arches, bridges and racecourses. Solved the way CBSE awards marks, with the key formulas and the mistakes that cost students marks every year, right on this page.
Conic Sections gets its name from the fact that all four curves — circle, parabola, ellipse, hyperbola — are what you get when a plane slices through a double cone at different angles. Each curve is then defined purely in terms of distances (from a point, or from a point and a line, or from two points), which is what lets every one of them be pinned down by a single clean equation once its centre or vertex is placed at the origin.
This Class 11 Maths NCERT Solutions Chapter 10 hub covers Conic Sections — the circle, parabola, ellipse and hyperbola — each defined as a locus satisfying a distance condition. A circle is every point at a fixed distance (the radius) from a fixed centre. A parabola is every point equidistant from a fixed point (the focus) and a fixed line (the directrix). An ellipse is every point where the sum of distances to two fixed foci stays constant, and a hyperbola is every point where the difference of those two distances stays constant.
Once the vertex or centre of each curve is placed at the origin with its axis along a coordinate axis, these geometric definitions collapse into clean standard equations: (x-h)^2+(y-k)^2=r^2 for the circle, four sign variations of y^2=4ax for the parabola, and \dfrac{x^2}{a^2}\pm\dfrac{y^2}{b^2}=1 for the ellipse and hyperbola (the ± being the only difference between them algebraically, despite very different shapes). The Miscellaneous Exercise closes the chapter by putting these same equations to work in physical settings — a satellite dish, an arch bridge, a suspension cable, a racecourse — where the real skill is recognising which conic a situation is describing.
Fixed distance from a centre — the simplest conic. Exercise 10.1.
Equidistant from a focus and a directrix — four standard orientations. Exercise 10.2.
Constant sum of distances to two foci — eccentricity always less than 1. Exercise 10.3.
Constant difference of distances to two foci — eccentricity always greater than 1. Exercise 10.4.
Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Conic Sections.
Centre (h, k), radius r. Every point on the circle is at distance r from the centre.
Complete the square in x and y to rewrite in standard form and read off (h, k) and r.
Vertex at origin, focus (a, 0), directrix x = −a. The other 3 orientations flip the sign or swap x and y.
The chord through the focus, perpendicular to the axis, with both ends on the parabola.
Major axis along whichever axis has the larger denominator (called a²); the smaller is b².
Foci lie inside the curve, at distance c from the centre; eccentricity is always less than 1.
The chord through either focus, perpendicular to the major axis, with both ends on the ellipse.
Transverse axis along whichever variable has the positive term.
Foci lie outside the curve; the "+" here (vs. the ellipse's "−") is what makes e always greater than 1.
Same formula as the ellipse, measured through either focus perpendicular to the transverse axis.
A quick way to decide, once you know what the question is actually describing.
Drawn from where students actually lose marks across all four exercises.
Circle — writing the equation from centre and radius, and recovering both by completing the square · 15 questions
Solve Exercise 10.1 →Parabola — focus, axis, directrix, latus rectum, and finding the equation from given conditions · 12 questions
Solve Exercise 10.2 →Ellipse — foci, vertices, axes, eccentricity, latus rectum, and finding the equation from given conditions · 20 questions
Solve Exercise 10.3 →Hyperbola — foci, vertices, eccentricity, latus rectum, and finding the equation from given conditions · 15 questions
Solve Exercise 10.4 →Real-world applications — reflectors, arches, bridge cables, sliding rods and racecourses · 8 questions
Solve Miscellaneous →Every formula for Conic Sections — plus every other Class 11 Maths chapter — in one printable PDF.
Get Formula Cards →Quick answers from Class 11 Maths NCERT Solutions Chapter 10, Conic Sections.
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