Conic Sections
24 solved problems from JEE previous year papers.
Hyperbola
JEE Main 2016 PYQ: find the eccentricity of a hyperbola whose latus rectum is 8 and whose conjugate axis equals half the distance…
Parabola
If y = mx + 4 Is a Tangent to Both the Parabolas y² = 4x and x² = 2by, Then b Is Equal To
JEE Main 2020 PYQ: the line y=mx+4 is tangent to both y²=4x and x²=2by. Find b by matching the tangency condition on the first…
Parabola
JEE Main 2019 PYQ: one end of a focal chord of the parabola y²=16x is at (1,4). Find the length of the chord by locating its…
Parabola
The normal at point P on the parabola y²=6x passes through (5,−8). If the tangent at P meets the directrix at Q, find the…
Ellipse
If the length of the minor axis of an ellipse equals half the distance between its foci, find the eccentricity using c=2b and…
Parabola
The locus of the midpoint of the segment joining the focus of y²=4ax to a moving point on the parabola is another parabola. Find…
Parabola
P(4,4√3) lies on y²=4ax, and PQ is a focal chord. M, N are feet of perpendiculars from P, Q to the directrix. Find the area of…
Parabola
Find point P on the parabola y²=8x at minimum distance from the centre C of x²+(y+6)²=1, then find the equation of the circle…
Ellipse
An ellipse centred at (1,0) has latus rectum 1/2 and its minor axis subtends 60° at the foci. Find the square of the sum of its…
Circle
Circle C passes through (4,0) and touches x²+y²+4x−6y=12 externally at (1,−1). Find C's radius using the…
Circle
Two circles in the first quadrant with radii r₁, r₂ touch both coordinate axes, each cutting a 2-unit intercept on x+y=2. Find…
Ellipse
The Equation of the Chord of the Ellipse x²/25 + y²/16 = 1 Whose Midpoint Is (3, 1) Is
Find the equation of the chord of the ellipse x²/25+y²/16=1 whose midpoint is (3,1), using the T=S₁ chord-of-midpoint formula.
Parabola
Shortest Distance Between Line y−x=1 and Curve x=y²
Find the shortest distance between the line y−x=1 and the parabola x=y² by parametrising a point on the curve and minimising the…
Circle
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 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
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
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
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
The Length of the Chord of the Ellipse x²/4 + y²/2 = 1 Whose Mid-Point Is (1, 1/2) Is
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…