Conic Sections

11 solved problems from JEE previous year papers.

Circle

Circle Touches x-axis at (a,0), y-axis Intercept b — Find (2a, b²)

A circle x²+y²−αx+βy+γ=0 touches the x-axis at (a,0) and cuts a y-axis intercept of length b, lying below the x-axis. Find the…
Circle

Three Circles of Radii a, b, c Touch Externally With x-axis as Common Tangent

Three circles of radii a, b, c touch each other externally with the x-axis as a common tangent. Derive the relation between a, b,…
Ellipse

Ellipse E and Hyperbola H Share Focal Distance 2√3, a−A=2 — Sum of Latus Rectums

Ellipse E and hyperbola H have the same distance between their foci (2√3), with a−A=2 and eccentricity ratio 1/3. Find the sum of…
Ellipse

αx+βy=109 Is Chord of x²/9+y²/4=1 With Midpoint (5/2,1/2), Find α+β

Given αx+βy=109 is the equation of a chord of x²/9+y²/4=1 with midpoint (5/2,1/2), find α+β using the T=S₁ chord-of-midpoint…
Parabola

Common Tangents to 4(x²+y²)=9 and y²=4x Meet at Q — Find l/e² of an Ellipse

The common tangents to 4(x²+y²)=9 and y²=4x meet at Q. An ellipse has semi-minor axis OQ and semi-major axis 6. Find l/e² for…
Circle

Circle Through A(2,−1), B(3,4), Center on (x−5)²+(y−1)²=13/2, Find r²

Circle C passes through A(2,−1) and B(3,4), AB not a diameter, and its centre lies on (x−5)²+(y−1)²=13/2. Find r² using the…
Hyperbola

Circle x²+y²=8x Meets Hyperbola x²/9−y²/4=1 at A,B — Centroid of PAB

A and B are intersection points of a circle and hyperbola; P moves on 2x−3y+4=0. Find the line on which the centroid of triangle…
Ellipse

Midpoint of a Chord of the Ellipse x²/9+y²/4=1 Is (√2, 4/3), Find α

If the midpoint of a chord of the ellipse x²/9+y²/4=1 is (√2, 4/3), and the chord's length is 2√α/3, then α is: (1) 18 (2) 22 (3)…
Ellipse

Ellipse E1 With Minor Axis Equal to Next Ellipse’s Major Axis, Sum of Areas

Let E1: x²/9+y²/4=1. Ellipses Ei share E1's centre and eccentricity, with minor axis of Ei equal to major axis of Ei+1. Find…
Ellipse

Two Ellipses, Eccentricity 1/√3: Area of Quadrilateral ABCD

Two ellipses with the same eccentricity, a given latus rectum product, and a known focal distance intersect at four points. A JEE…
Ellipse

Chord Length of Ellipse x²/4+y²/2=1 With Midpoint (1,½)

Given only the midpoint of a chord of an ellipse, find its length using the T = S1 equation of a chord — a JEE Main 2025 conic…