LFM Pure

jee-main 2021 Q69 Circle-Conic Interaction with Tangency or Intersection View

If a tangent to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $B C$ as diameter passes through the point.
(1) $( \sqrt { 3 } , 0 )$
(2) $( \sqrt { 2 } , 0 )$
(3) $( 1,1 )$
(4) $( - 1,1 )$

jee-main 2021 Q70 Circle Identification and Classification View

Choose the correct statement about two circles whose equations are given below: $x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + 41 = 0$ $x ^ { 2 } + y ^ { 2 } - 22 x - 10 y + 137 = 0$
(1) circles have same centre
(2) circles have no meeting point

jee-main 2021 Q83 Tangent Lines and Tangent Lengths View

A line is a common tangent to the circle $( x - 3 ) ^ { 2 } + y ^ { 2 } = 9$ and the parabola $y ^ { 2 } = 4 x$. If the two points of contact $( a , b )$ and $( c , d )$ are distinct and lie in the first quadrant, then $2 ( a + c )$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2021 Q83 Tangent Lines and Tangent Lengths View

Let $A B C D$ be a square of side of unit length. Let a circle $C _ { 1 }$ centered at $A$ with unit radius is drawn. Another circle $C _ { 2 }$ which touches $C _ { 1 }$ and the lines $A D$ and $A B$ are tangent to it, is also drawn. Let a tangent line from the point $C$ to the circle $C _ { 2 }$ meet the side $A B$ at $E$. If the length of $E B$ is $\alpha + \sqrt { 3 } \beta$, where $\alpha , \beta$ are integers, then $\alpha + \beta$ is equal to $\_\_\_\_$.

jee-main 2021 Q84 Circle-Line Intersection and Point Conditions View

If the variable line $3 x + 4 y = \alpha$ lies between the two circles $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$ and $( x - 9 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is

jee-main 2021 Q85 Geometric or applied optimisation problem View

If the point on the curve $y ^ { 2 } = 6 x$, nearest to the point $\left( 3 , \frac { 3 } { 2 } \right)$ is $( \alpha , \beta )$, then $2 ( \alpha + \beta )$ is equal to $\underline{\hspace{1cm}}$.

jee-main 2021 Q85 Circles Tangent to Each Other or to Axes View

Two circles each of radius 5 units touch each other at the point $( 1,2 )$. If the equation of their common tangent is $4 x + 3 y = 10$, and $C _ { 1 } ( \alpha , \beta )$ and $C _ { 2 } ( \gamma , \delta ) , C _ { 1 } \neq C _ { 2 }$ are their centres, then $| ( \alpha + \beta ) ( \gamma + \delta ) |$ is equal to

jee-main 2022 Q63 Circle Equation Derivation View

Let a circle $C$ touch the lines $L _ { 1 } : 4 x - 3 y + K _ { 1 } = 0$ and $L _ { 2 } : 4 x - 3 y + K _ { 2 } = 0 , K _ { 1 } , \quad K _ { 2 } \in R$. If a line passing through the centre of the circle $C$ intersects $L _ { 1 }$ at $( -1, 2 )$ and $L _ { 2 }$ at $( 3 , - 6 )$, then the equation of the circle $C$ is
(1) $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$
(2) $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
(3) $( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$
(4) $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 16$

jee-main 2022 Q63 Tangent and Normal Line Problems View

If $m$ is the slope of a common tangent to the curves $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 12$, then $12 \mathrm {~m} ^ { 2 }$ is equal to
(1) 6
(2) 9
(3) 10
(4) 12

jee-main 2022 Q63 Area and Geometric Measurement Involving Circles View

Let the tangents at two points $A$ and $B$ on the circle $x ^ { 2 } + y ^ { 2 } - 4 x + 3 = 0$ meet at origin $O ( 0,0 )$. Then the area of the triangle of $O A B$ is
(1) $\frac { 3 \sqrt { 3 } } { 2 }$
(2) $\frac { 3 \sqrt { 3 } } { 4 }$
(3) $\frac { 3 } { 2 \sqrt { 3 } }$
(4) $\frac { 3 } { 4 \sqrt { 3 } }$

jee-main 2022 Q64 Tangent Lines and Tangent Lengths View

If $y = m _ { 1 } x + c _ { 1 }$ and $y = m _ { 2 } x + c _ { 2 } , \quad m _ { 1 } \neq m _ { 2 }$ are two common tangents of circle $x ^ { 2 } + y ^ { 2 } = 2$ and parabola $y ^ { 2 } = x$, then the value of $8 \quad m _ { 1 } \quad m _ { 2 }$ is equal to
(1) $3 \sqrt { 2 } - 4$
(2) $6 \sqrt { 2 } - 4$
(3) $- 5 + 6 \sqrt { 2 }$
(4) $3 + 4 \sqrt { 2 }$

jee-main 2022 Q64 Locus and Trajectory Derivation View

The locus of the mid-point of the line segment joining the point $( 4,3 )$ and the points on the ellipse $x ^ { 2 } + 2 y ^ { 2 } = 4$ is an ellipse with eccentricity
(1) $\frac { \sqrt { 3 } } { 2 }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } }$
(4) $\frac { 1 } { 2 }$

jee-main 2022 Q64 Area and Geometric Measurement Involving Circles View

If the tangents drawn at the point $O(0,0)$ and $P(1+\sqrt{5}, 2)$ on the circle $x^2 + y^2 - 2x - 4y = 0$ intersect at the point $Q$, then the area of the triangle $OPQ$ is equal to
(1) $\frac{3+\sqrt{5}}{2}$
(2) $\frac{4+2\sqrt{5}}{2}$
(3) $\frac{5+3\sqrt{5}}{2}$
(4) $\frac{7+3\sqrt{5}}{2}$

jee-main 2022 Q64 Circle Equation Derivation View

Let the abscissae of the two points $P$ and $Q$ on a circle be the roots of $x ^ { 2 } - 4 x - 6 = 0$ and the ordinates of $P$ and $Q$ be the roots of $y ^ { 2 } + 2 y - 7 = 0$. If $PQ$ is a diameter of the circle $x ^ { 2 } + y ^ { 2 } + 2 a x + 2 b y + c = 0$, then the value of $a + b - c$ is
(1) 12
(2) 13
(3) 14
(4) 16

jee-main 2022 Q64 Circle Equation Derivation View

Let $C$ be a circle passing through the points $A ( 2 , - 1 )$ and $B ( 3,4 )$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $( x - 5 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 13 } { 2 }$, then $r ^ { 2 }$ is equal to
(1) 32
(2) $\frac { 65 } { 2 }$
(3) $\frac { 61 } { 2 }$
(4) 30

jee-main 2022 Q64 Area and Geometric Measurement Involving Circles View

A point $P$ moves so that the sum of squares of its distances from the points $( 1,2 )$ and $( - 2,1 )$ is 14. Let $f ( x , y ) = 0$ be the locus of $P$, which intersects the $x$-axis at the points $A , B$ and the $y$-axis at the point $C , D$. Then the area of the quadrilateral $ACBD$ is equal to
(1) $\frac { 9 } { 2 }$
(2) $\frac { 3 \sqrt { 17 } } { 2 }$
(3) $\frac { 3 \sqrt { 17 } } { 4 }$
(4) 9

jee-main 2022 Q65 Circle-Related Locus Problems View

A particle is moving in the $x y$-plane along a curve $C$ passing through the point $( 3,3 )$. The tangent to the curve $C$ at the point $P$ meets the $x$-axis at $Q$. If the $y$-axis bisects the segment $P Q$, then $C$ is a parabola with
(1) length of latus rectum 3
(2) length of latus rectum 6
(3) focus $\left( \frac { 4 } { 3 } , 0 \right)$
(4) focus $\left( 0 , \frac { 3 } { 3 } \right)$

jee-main 2022 Q65 Tangent and Normal Line Problems View

The normal to the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 9 } = 1$ at the point $( 8,3 \sqrt { 3 } )$ on it passes through the point
(1) $( 15 , - 2 \sqrt { 3 } )$
(2) $( 9,2 \sqrt { 3 } )$
(3) $( - 1,9 \sqrt { 3 } )$
(4) $( - 1,6 \sqrt { 3 } )$

jee-main 2022 Q65 Area and Geometric Measurement Involving Circles View

Let the tangent to the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 2$ at the point $M ( - 1,1 )$ intersect the circle $C _ { 2 }$ : $( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 5$, at two distinct points $A$ and $B$. If the tangents to $C _ { 2 }$ at the points $A$ and $B$ intersect at $N$, then the area of the triangle $A N B$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 5 } { 3 }$

jee-main 2022 Q65 Triangle Properties and Special Points View

For $t \in (0, 2\pi)$, if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$, $B(\cos t, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $a^2 - b^2$ is equal to
(1) $\frac{8}{3}$
(2) $8$
(3) $\frac{77}{9}$
(4) $\frac{80}{9}$

jee-main 2022 Q65 Tangent Lines and Tangent Lengths View

Let the normal at the point $P$ on the parabola $y ^ { 2 } = 6 x$ pass through the point $( 5 , - 8 )$. If the tangent at $P$ to the parabola intersects its directrix at the point $Q$, then the ordinate of the point $Q$ is
(1) $\frac { - 9 } { 4 }$
(2) $\frac { 9 } { 4 }$
(3) $\frac { - 5 } { 2 }$
(4) $- 3$

jee-main 2022 Q65 Circle-Related Locus Problems View

Let the locus of the centre $(\alpha , \beta),\ \beta > 0$, of the circle which touches the circle $x ^ { 2 } + (y - 1) ^ { 2 } = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is
(1) $\frac { 32 \sqrt { 2 } } { 3 }$
(2) $\frac { 40 \sqrt { 2 } } { 3 }$
(3) $\frac { 64 } { 3 }$
(4) $\frac { 32 } { 3 }$

jee-main 2022 Q65 Tangent and Normal Line Problems View

Let the tangent drawn to the parabola $y ^ { 2 } = 24 x$ at the point $( \alpha , \beta )$ is perpendicular to the line $2 x + 2 y = 5$. Then the normal to the hyperbola $\frac { x ^ { 2 } } { \alpha ^ { 2 } } - \frac { y ^ { 2 } } { \beta ^ { 2 } } = 1$ at the point $( \alpha + 4 , \beta + 4 )$ does NOT pass through the point:
(1) $( 25,10 )$
(2) $( 20,12 )$
(3) $( 30,8 )$
(4) $( 15,13 )$

jee-main 2022 Q65 Chord Length and Chord Properties View

If the circle $x ^ { 2 } + y ^ { 2 } - 2 g x + 6 y - 19 c = 0 , g , c \in \mathbb { R }$ passes through the point $( 6,1 )$ and its centre lies on the line $x - 2 c y = 8$, then the length of intercept made by the circle on $x$-axis is
(1) $\sqrt { 11 }$
(2) 4
(3) 3
(4) $2 \sqrt { 23 }$

jee-main 2022 Q66 Circle Equation Derivation View

Let $x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$ be a circle passing through ( 0,6 ) and touching the parabola $y = x ^ { 2 }$ at ( 2,4 ). Then $A + C$ is equal to
(1) 16
(2) $\frac { 88 } { 5 }$
(3) 72
(4) - 8

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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