LFM Pure

jee-main 2023 Q67 Circle Equation Derivation View

Let the centre of a circle $C$ be $( \alpha , \beta )$ and its radius $r < 8$. Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to $C$. Then $( \alpha - \beta + r )$ is equal to
(1) 7
(2) 5
(3) 6
(4) 9

jee-main 2023 Q67 Tangent Lines and Tangent Lengths View

The number of common tangents, to the circles $x ^ { 2 } + y ^ { 2 } - 18 x - 15 y + 131 = 0$ and $x ^ { 2 } + y ^ { 2 } - 6 x - 6 y - 7 = 0$, is
(1) 3
(2) 1
(3) 4
(4) 2

jee-main 2023 Q68 Chord Length and Chord Properties View

The set of all values of $a^2$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$, is equal to:
(1) $(8, \infty)$
(2) $(0, 4]$
(3) $(4, \infty)$
(4) $(2, 12]$

jee-main 2023 Q68 Circle-Conic Interaction with Tangency or Intersection View

Let a circle of radius 4 be concentric to the ellipse $15 x ^ { 2 } + 19 y ^ { 2 } = 285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 12 }$

jee-main 2023 Q68 Circle Equation Derivation View

The points of intersection of the line $a x + b y = 0 , ( \mathrm { a } \neq \mathrm { b } )$ and the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } - 2 \mathrm { x } = 0$ are $A ( \alpha , 0 )$ and $B ( 1 , \beta )$. The image of the circle with $A B$ as a diameter in the line $\mathrm { x } + \mathrm { y } + 2 = 0$ is:
(1) $x ^ { 2 } + y ^ { 2 } + 5 x + 5 y + 12 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + 5 y + 8 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + 3 y + 4 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 5 x - 5 y + 12 = 0$

jee-main 2023 Q68 Area and Geometric Measurement Involving Circles View

Let $P(a_{1}, b_{1})$ and $Q(a_{2}, b_{2})$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}$ is equal to $\_\_\_\_$

jee-main 2023 Q68 Locus Determination View

If the point $\left( \alpha , \frac { 7 \sqrt { 3 } } { 3 } \right)$ lies on the curve traced by the mid-points of the line segments of the lines $x \cos \theta + y \sin \theta = 7 , \theta \in \left( 0 , \frac { \pi } { 2 } \right)$ between the co-ordinates axes, then $\alpha$ is equal to
(1) - 7
(2) $- 7 \sqrt { 3 }$
(3) $7 \sqrt { 3 }$
(4) 7

jee-main 2023 Q69 Circle-Related Locus Problems View

The locus of the middle points of the chords of the circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 4$ which subtend an angle $\theta _ { i }$ at the centre of the circle $C _ { i }$, is a circle of radius $r _ { i }$. If $\theta _ { 1 } = \frac { \pi } { 3 } , \theta _ { 3 } = \frac { 2 \pi } { 3 }$ and $r _ { 1 } ^ { 2 } = r _ { 2 } ^ { 2 } + r _ { 3 } ^ { 2 }$, then $\theta _ { 2 }$ is equal to
(1) $\frac { \pi } { 4 }$
(2) $\frac { 3 \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 2 }$

jee-main 2023 Q69 Geometric or applied optimisation problem View

Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P(b, c)$, $b, c \in N$, on the parabola $y^2 = 2ax$ and the lines $x = b$, $y = 0$ is 16 unit$^2$, then $\sum_{a \in S} a$ is equal to $\_\_\_\_$.

jee-main 2023 Q69 Triangle Properties and Special Points View

Let $A ( 0,1 ) , B ( 1,1 )$ and $C ( 1,0 )$ be the mid-points of the sides of a triangle with incentre at the point $D$. If the focus of the parabola $y ^ { 2 } = 4 a x$ passing through $D$ is $( \alpha + \beta \sqrt { 2 } , 0 )$, where $\alpha$ and $\beta$ are rational numbers, then $\frac { \alpha } { \beta ^ { 2 } }$ is equal to
(1) 8
(2) 12
(3) 6
(4) $\frac { 9 } { 2 }$

jee-main 2023 Q69 Triangle Properties and Special Points View

Let $C ( \alpha , \beta )$ be the circumcentre of the triangle formed by the lines $4 x + 3 y = 69$, $4 y - 3 x = 17$, and $x + 7 y = 61$. Then $( \alpha - \beta ) ^ { 2 } + \alpha + \beta$ is equal to
(1) 18
(2) 17
(3) 15
(4) 16

jee-main 2023 Q70 Area and Geometric Measurement Involving Circles View

If the tangents at the points $P$ and $Q$ on the circle $x^{2} + y^{2} - 2x + y = 5$ meet at the point $R\left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is
(1) $\frac{5}{4}$
(2) $\frac{13}{8}$
(3) $\frac{5}{8}$
(4) $\frac{13}{4}$

jee-main 2023 Q70 Inscribed/Circumscribed Circle Computations View

Let $O$ be the origin and $O P$ and $O Q$ be the tangents to the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 8 = 0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $O P Q$ passes through the point $\left( \alpha , \frac { 1 } { 2 } \right)$, then a value of $\alpha$ is
(1) $\frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) 1

jee-main 2023 Q70 Tangent Lines and Tangent Lengths View

A circle with centre $( 2,3 )$ and radius 4 intersects the line $x + y = 3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S ( \alpha , \beta )$, then $4 \alpha - 7 \beta$ is equal to $\_\_\_\_$

jee-main 2023 Q70 Tangent Lines and Tangent Lengths View

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR)^{2}$ is equal to
(1) 64
(2) 76
(3) 81
(4) 72

jee-main 2023 Q70 Circle Equation Derivation View

Consider a circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = \alpha - 5$. Let its mirror image in the line $y = 2 x + 1$ be another circle $C _ { 2 } : 5 x ^ { 2 } + 5 y ^ { 2 } - 10 f x - 10 g y + 36 = 0$. Let $r$ be the radius of $C _ { 2 }$. Then $\alpha + r$ is equal to $\_\_\_\_$

jee-main 2023 Q70 Circles Tangent to Each Other or to Axes View

Two circles in the first quadrant of radii $r _ { 1 }$ and $r _ { 2 }$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$. Then $r _ { 1 } { } ^ { 2 } + r _ { 2 } { } ^ { 2 } - r _ { 1 } r _ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2023 Q71 Circle-Related Locus Problems View

The ordinates of the points $P$ and $Q$ on the parabola with focus $( 3,0 )$ and directrix $x = - 3$ are in the ratio 3:1. If $R ( \alpha , \beta )$ is the point of intersection of the tangents to the parabola at $P$ and $Q$, then $\frac { \beta ^ { 2 } } { \alpha }$ is equal to

jee-main 2023 Q71 Tangent Lines and Tangent Lengths View

Points $P ( - 3,2 ) , Q ( 9,10 )$ and $R ( \alpha , 4 )$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - k y = 1$, then $k$ is equal to $\_\_\_\_$.

jee-main 2023 Q71 Tangent Lines and Tangent Lengths View

Let the tangents at the points $A ( 4 , - 11 )$ and $B ( 8 , - 5 )$ on the circle $x ^ { 2 } + y ^ { 2 } - 3 x + 10 y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
(1) $\frac { 3 \sqrt { 3 } } { 4 }$
(2) $2 \sqrt { 13 }$
(3) $\sqrt { 13 }$
(4) $\frac { 2 \sqrt { 13 } } { 3 }$

jee-main 2023 Q71 Inscribed/Circumscribed Circle Computations View

A triangle is formed by the tangents at the point $( 2,2 )$ on the curves $y ^ { 2 } = 2 x$ and $x ^ { 2 } + y ^ { 2 } = 4 x$, and the line $\mathrm { x } + \mathrm { y } + 2 = 0$. If $r$ is the radius of its circumcircle, then $r ^ { 2 }$ is equal to $\_\_\_\_$

jee-main 2023 Q75 Circle Identification and Classification View

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^{2}x^{2} + \alpha^{2}y^{2}\right) = \alpha^{2}\beta^{2}$ is
(1) $\frac{\sqrt{119}}{12}$
(2) $\frac{\sqrt{117}}{12}$
(3) $\frac{3\sqrt{15}}{12}$
(4) $\frac{\sqrt{129}}{12}$

jee-main 2023 Q78 Circle-Line Intersection and Point Conditions View

The number of integral values of $k$ for which the line $3x + 4y = k$ intersects the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ at two distinct points is $\_\_\_\_$.

jee-main 2023 Q85 Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

The coefficient of $x^7$ in $(1 - x + 2x^3)^{10}$ is $\_\_\_\_$.

jee-main 2024 Q65 Intersection of Circles or Circle with Conic View

If the circles $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 4 = 0$ intersect at exactly two distinct points, then
(1) $5 < \mathrm { r } < 9$
(2) $0 < \mathrm { r } < 7$
(3) $3 < r < 7$
(4) $\frac { 1 } { 2 } < \mathrm { r } < 7$

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164