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 ordered pair (2a, b²).
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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, c using centre coordinates and distance.
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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 their latus rectum lengths.
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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 formula.
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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 this ellipse.
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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 perpendicular-bisector property.
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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 PAB always lies.
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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) 26 (4) 20.
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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 (5/π)ΣAi.
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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 Main 2025 area problem, solved.