Conic Sections · Question 88

JEE Main 2025 (Online) 28th January Morning ShiftMCQMedium📈 Topic in 26% papers+4−1

Let the equation of the circle, which touches x-axis at the point (a, 0), a > 0 and cuts off an intercept of length b on y-axis be x2 + y2 - x + y + = 0. If the circle lies below x-axis, then the ordered pair (2a, b2) is equal to

( , 2 + 4 )
( , 2 - 4 )
( , 2 - 4 )
( , 2 + 4 )

Answer

(2) ( , 2 - 4 )

Solution

The circle touches the x-axis at (a, 0) and lies below it centre (a, -r), radius r (r > 0).

Equation: (x - a)2 + (y + r)2 = r2

x2 + y2 - 2ax + 2ry + a2 = 0

Comparing with x2 + y2 - x + y + = 0:

= 2a, = 2r, = a2

Intercept on y-axis: put x = 0 y2 + 2ry + a2 = 0

b = |y1 - y2| = 2 r2 - a2

b2 = 4r2 - 4a2 = 2 - 4

(2a, b2) = ( , 2 - 4 )

Answer: (2)

JEE Main 2025 (Online) 28th January Morning Shift

Keep practising — Conic Sections

Let E1 : x29 + y24 = 1 be an ellipse. Ellipses Ei's are constructed such that their centres and eccentricities are same as that of E1, and the length of minor axis of Ei is the length of major axis of Ei+1 (i 1). If Ai is the area of the ellipse Ei, then 5 ( i=1 Ai ), is equal to .

JEE Main 2025 (Online) 28th January Morning Shift

If the midpoint of a chord of the ellipse x29 + y24 = 1 is ( 2, 4/3), and the length of the chord is 2 3, then is:

JEE Main 2025 (Online) 28th January Evening Shift

If A and B are the points of intersection of the circle x2 + y2 - 8x = 0 and the hyperbola x29 - y24 = 1 and a point P moves on the line 2x - 3y + 4 = 0, then the centroid of PAB lies on the line:

x + 9y = 36
9x - 9y = 32
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JEE Main 2025 (Online) 28th January Evening Shift

Let A and B be the two points of intersection of the line y + 5 = 0 and the mirror image of the parabola y2 = 4x with respect to the line x + y + 4 = 0. If d denotes the distance between A and B, and a denotes the area of SAB, where S is the focus of the parabola y2 = 4x, then the value of (a + d) is .

JEE Main 2025 (Online) 28th January Evening Shift

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