LFM Pure

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 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 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 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 Tangent Lines and Tangent Lengths View

The equation of a common tangent to the parabolas $y = x ^ { 2 }$ and $y = -(x - 2) ^ { 2 }$ is
(1) $y = 4 x - 2$
(2) $y = 4 x - 1$
(3) $y = 4 x + 1$
(4) $y = 4 x + 2$

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 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

jee-main 2022 Q66 Area and Geometric Measurement Involving Circles View

The tangents at the points $A(1,3)$ and $B(1,-1)$ on the parabola $y^2 - 2x - 2y = 1$ meet at the point $P$. Then the area (in unit$^2$) of the triangle $PAB$ is:
(1) 4
(2) 6
(3) 7
(4) 8

jee-main 2022 Q66 Area and Geometric Measurement Involving Circles View

Let $C$ be the centre of the circle $x^2 + y^2 - x + 2y = \frac{11}{4}$ and $P$ be a point on the circle. A line passes through the point $C$, makes an angle of $\frac{\pi}{4}$ with the line $CP$ and intersects the circle at the points $Q$ and $R$. Then the area of the triangle $PQR$ (in unit$^2$) is
(1) 2
(2) $2\sqrt{2}$
(3) $8\sin\frac{\pi}{8}$
(4) $8\cos\frac{\pi}{8}$

jee-main 2022 Q66 Circle Identification and Classification View

The set of values of $k$ for which the circle $C : 4 x ^ { 2 } + 4 y ^ { 2 } - 12 x + 8 y + k = 0$ lies inside the fourth quadrant and the point $\left( 1 , - \frac { 1 } { 3 } \right)$ lies on or inside the circle $C$ is
(1) An empty set
(2) $\left( 6 , \frac { 95 } { 9 } \right]$
(3) $\left[ \frac { 80 } { 9 } , 10 \right)$
(4) $\left( 9 , \frac { 92 } { 9 } \right]$

jee-main 2022 Q66 Tangent Lines and Tangent Lengths View

Let $P ( a , b )$ be a point on the parabola $y ^ { 2 } = 8 x$ such that the tangent at $P$ passes through the centre of the circle $x ^ { 2 } + y ^ { 2 } - 10 x - 14 y + 65 = 0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of all possible values of $b$. Then the value of $A + B$ is equal to
(1) 0
(2) 25
(3) 40
(4) 65

jee-main 2022 Q66 Tangent Lines and Tangent Lengths View

A circle $C _ { 1 }$ passes through the origin $O$ and has diameter 4 on the positive $x$-axis. The line $y = 2 x$ gives a chord $O A$ of a circle $C _ { 1 }$. Let $C _ { 2 }$ be the circle with $O A$ as a diameter. If the tangent to $C _ { 2 }$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $Q A : A P$ is equal to
(1) $1 : 4$
(2) $1 : 5$
(3) $2 : 5$
(4) $1 : 3$

jee-main 2022 Q67 Circle-Related Locus Problems View

A circle touches both the $y$-axis and the line $x + y = 0$. Then the locus of its center is
(1) $y = \sqrt{2}x$
(2) $x = \sqrt{2}y$
(3) $y^2 - x^2 = 2xy$
(4) $x^2 - y^2 = 2xy$

jee-main 2022 Q67 Inscribed/Circumscribed Circle Computations View

Let $A(\alpha, -2)$, $B(\alpha, 6)$ and $C\left(\frac{\alpha}{4}, -2\right)$ be vertices of a $\triangle ABC$. If $\left(5, \frac{\alpha}{4}\right)$ is the circumcentre of $\triangle ABC$, then which of the following is NOT correct about $\triangle ABC$
(1) area is 24
(2) perimeter is 25
(3) circumradius is 5
(4) inradius is 2

jee-main 2022 Q67 Circle Equation Derivation View

If the length of the latus rectum of a parabola, whose focus is $( a , a )$ and the tangent at its vertex is $x + y = a$, is 16 , then $| a |$ is equal to
(1) $2 \sqrt { 2 }$
(2) $2 \sqrt { 3 }$
(3) $4 \sqrt { 2 }$
(4) 4

jee-main 2022 Q68 Chord Length and Chord Properties View

The line $y = x + 1$ meets the ellipse $\frac{x^2}{4} + \frac{y^2}{2} = 1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $3r^2$ is equal to
(1) 20
(2) 12
(3) 11
(4) 8

jee-main 2022 Q73 Optimization on a Circle View

Let $P$ and $Q$ be any points on the curves $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$ and $y = x ^ { 2 }$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval
(1) $\left( 0 , \frac { 1 } { 4 } \right)$
(2) $\left( \frac { 1 } { 2 } , \frac { 3 } { 4 } \right)$
(3) $\left( \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 3 } { 4 } , 1 \right)$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144