Two Dimensional Geometry jee mains questions | JEE Main Two Dimensional Geometry Previous Year Question

Q 51 :

Let the set of all values of $r$ , for which the circles ${(x + 1)}^{2} + {(y + 4)}^{2} = r^{2} and x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to [2026]

24
21
20
25

1

2

3

4

(4)

${(x - 2)}^{2} + {(y - 1)}^{2} = 3^{2} & {(x + 1)}^{2} + {(y + 4)}^{2} = r^{2}$

$| r_{1} - r_{2} |$ $< c_{1} c_{2} < r_{1} + r_{2}$

$| r - 3 |$ $< \sqrt{{(2 + 1)}^{2} + {(1 + 4)}^{2}} < r + 3$

$| r - 3 |$ $< \sqrt{34} & r + 3 > \sqrt{34}$

$- \sqrt{34} < r - 3 < \sqrt{34} & r > \sqrt{34} - 3$

$i.e. r = (3 - \sqrt{34}, 3 + \sqrt{34}) \cap (\sqrt{34} - 3, \infty)$

$i.e. r \in (\sqrt{34} - 3, \sqrt{34} + 3)$

$∴$ $α β = (\sqrt{34} - 3) (\sqrt{34} + 3)$

$= 34 - 9$

$= 25$

Q 52 :

Let the circle $x^{2} + y^{2} = 4$ intersect x-axis at the points $A (a, 0), a > 0 a n d B (b, 0) .$ Let $P (2 \cos α, 2 \sin α) 0 < α < \frac{π}{2} a n d Q (2 \cos β, 2 \sin β)$ be two points such that $(α - β) = \frac{π}{2}$ . Then the point of intersection of AQ and BP lies on : [2026]

$x^{2} + y^{2} - 4 x - 4 = 0$
$x^{2} + y^{2} - 4 y - 4 = 0$
$x^{2} + y^{2} - 4 x - 4 y - 4 = 0$
$x^{2} + y^{2} - 4 x - 4 y = 0$

1

2

3

4

(2)

Q 53 :

Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

0%

(9)

Q 54 :

If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

0%

(2)

Mid point of $P Q$ lies on $x - y + 1 = 0$

$\frac{2 \cos θ + α}{2} - \frac{2 \sin θ - 5 α - 2}{2} + 1 = 0$

$2 \cos θ + α - 2 \sin θ + 5 α + 2 + 2 = 0$

$\cos θ - \sin θ + 3 α + 2 = 0 . . . (1)$

$∵$ $Slope of P Q is - 1$

$\frac{2 \sin θ + 5 α + 2}{2 \cos θ - α} = - 1$

$2 \sin θ + 5 α + 2 = - 2 \cos θ + α$

$\sin θ + \cos θ + 2 α + 1 = 0 . . . (2)$

Eliminate $α$ from (1) and (2)

$\Rightarrow \cos θ + 5 \sin θ = 1, θ \in [0, 2 π]$

$\Rightarrow 5 \times 2 \sin \frac{θ}{2} \cos \frac{θ}{2} = 2 \sin^{2} \frac{θ}{2}$

$∴$ $\sin \frac{θ}{2} = 0 \Rightarrow \cos θ = 1$

or

$\sin \frac{θ}{2} = 5 \Rightarrow \cos θ = - \frac{12}{13}$

Sum of all possible values of abscissa of point $P$ is

$= 2 \times 1 + 2 (\frac{- 12}{13}) = \frac{2}{13}$

$∴$ $13 times sum of all possible values of abscissa of point P is = 2$

Q 55 :

Let y=x be the equation of a chord of the circle $C_{1}$ (in the closed half-plane $x \geq 0$ ) of diameter 10 passing through the origin. Let $C_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $C_{2}$ , which passes through the point (2,3) and is farthest from the center of $C_{2}$ , is $x + a y + b = 0$ ,

then a−b is equal to [2026]

10
-2
6
-6

1

2

3

4

(2)

Topic Question Set

Q 51 :

Let the set of all values of $r$ , for which the circles ${(x + 1)}^{2} + {(y + 4)}^{2} = r^{2} and x^{2} + y^{2} - 4 x - 2 y - 4 = 0$ intersect at two distinct points be the interval $(α, β)$ . Then $α β$ is equal to [2026]

Q 53 :

Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

0%

Q 54 :

If P is a point on the circle $x^{2} + y^{2} = 4,$ , Q is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of PQ, then 13 times the sum of abscissa of all such points P is __________. [2026]

0%

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