LFM Pure

jee-main 2016 Q74 Circle Equation Derivation View

If one of the diameters of the circle, given by the equation, $x^2 + y^2 - 4x + 6y - 12 = 0$, is a chord of a circle $S$, whose centre is at $(-3, 2)$, then the radius of $S$ is:
(1) $5\sqrt{2}$
(2) $5\sqrt{3}$
(3) $5$
(4) $10$

jee-main 2016 Q75 Circle Equation Derivation View

Let $P$ be the point on the parabola, $y^2 = 8x$, which is at a minimum distance from the centre $C$ of the circle, $x^2 + (y+6)^2 = 1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is:
(1) $x^2 + y^2 - 4x + 8y + 12 = 0$
(2) $x^2 + y^2 - x + 4y - 12 = 0$
(3) $x^2 + y^2 - \frac{x}{4} + 2y - 24 = 0$
(4) $x^2 + y^2 - 4x + 9y + 18 = 0$

jee-main 2017 Q63 Circles Tangent to Each Other or to Axes View

The radius of a circle, having minimum area, which touches the curve $y = 4 - x ^ { 2 }$ and the lines $y = | x |$ is:
(1) $2 ( \sqrt { 2 } + 1 )$
(2) $2 ( \sqrt { 2 } - 1 )$
(3) $4 ( \sqrt { 2 } - 1 )$
(4) $4 ( \sqrt { 2 } + 1 )$

jee-main 2017 Q67 Circle-Related Locus Problems View

The locus of the point of intersection of the straight lines, $t x - 2 y - 3 t = 0$ and $x - 2 t y + 3 = 0 ( t \in R )$, is:
(1) A hyperbola with the length of conjugate axis 3
(2) A hyperbola with eccentricity $\sqrt { 5 }$
(3) An ellipse with the length of major axis 6
(4) An ellipse with eccentricity $\frac { 2 } { \sqrt { 5 } }$

jee-main 2017 Q68 Optimization on a Circle View

The radius of a circle, having minimum area, which touches the curve $y = 4 - x^2$ and the lines $y = |x|$ is:
(1) $2(\sqrt{2} + 1)$
(2) $2(\sqrt{2} - 1)$
(3) $4(\sqrt{2} - 1)$
(4) $4(\sqrt{2} + 1)$

jee-main 2017 Q68 Chord Length and Chord Properties View

If two parallel chords of a circle, having diameter 4 units, lie on the opposite sides of the center and subtend angles $\cos ^ { - 1 } \left( \frac { 1 } { 7 } \right)$ and $\sec ^ { - 1 } ( 7 )$ at the center respectively, then the distance between these chords is:
(1) $\frac { 8 } { \sqrt { 7 } }$
(2) $\frac { 16 } { 7 }$
(3) $\frac { 4 } { \sqrt { 7 } }$
(4) $\frac { 8 } { 7 }$

jee-main 2017 Q69 Tangent Lines and Tangent Lengths View

If the common tangents to the parabola, $x ^ { 2 } = 4 y$ and the circle, $x ^ { 2 } + y ^ { 2 } = 4$ intersect at the point $P$, then the distance of $P$ from the origin (units), is:
(1) $2 ( 3 + 2 \sqrt { 2 } )$
(2) $3 + 2 \sqrt { 2 }$
(3) $\sqrt { 2 } + 1$
(4) $2 ( \sqrt { 2 } + 1 )$

jee-main 2017 Q70 Optimization on a Circle View

If a point $P ( 0 , - 2 )$ and $Q$ is any point on the circle, $x ^ { 2 } + y ^ { 2 } - 5 x - y + 5 = 0$, then the maximum value of $( P Q ) ^ { 2 }$ is
(1) $8 + 5 \sqrt { 3 }$
(2) $\frac { 47 + 10 \sqrt { 6 } } { 2 }$
(3) $14 + 5 \sqrt { 3 }$
(4) $\frac { 25 + \sqrt { 6 } } { 2 }$

jee-main 2017 Q71 Area and Geometric Measurement Involving Circles View

Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac { 3 } { 5 }$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilateral inscribed in the ellipse, with the vertices as the vertices of the ellipse, is:
(1) 32
(2) 80
(3) 40
(4) 8

jee-main 2018 Q68 Circle Equation Derivation View

A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line $y - 4 x + 3 = 0$, then its radius is equal to :
(1) $\sqrt { 5 }$
(2) $\sqrt { 2 }$
(3) 2
(4) 1

jee-main 2018 Q69 Circles Tangent to Each Other or to Axes View

If a circle $C$, whose radius is 3 , touches externally the circle $x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 4 = 0$ at the point $( 2,2 )$, then the length of the intercept cut by this circle $C$ on the $x$-axis is equal to
(1) $2 \sqrt { 3 }$
(2) $\sqrt { 5 }$
(3) $3 \sqrt { 2 }$
(4) $2 \sqrt { 5 }$

jee-main 2018 Q69 Tangent Lines and Tangent Lengths View

If the tangent at $( 1,7 )$ to the curve $x ^ { 2 } = y - 6$ touch the circle $x ^ { 2 } + y ^ { 2 } + 16 x + 12 y + c = 0$ then the value of $c$ is:
(1) 95
(2) 195
(3) 185
(4) 85

jee-main 2018 Q69 Tangent Lines and Tangent Lengths View

Two parabolas with a common vertex and with axes along the $x$-axis and $y$-axis respectively, intersect each other in the first quadrant. If the length of the latus rectum of each parabola is 3, then the equation of the common tangent to the two parabolas is :
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $x + 2 y + 3 = 0$
(4) $4 ( x + y ) + 3 = 0$

jee-main 2018 Q69 Circle Equation Derivation View

A circle passes through the points $( 2,3 )$ and $( 4,5 )$. If its centre lies on the line, $y - 4 x + 3 = 0$, then its radius is equal to
(1) $\sqrt { 5 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2

jee-main 2018 Q70 Inscribed/Circumscribed Circle Computations View

Tangent and normal are drawn at $P ( 16,16 )$ on the parabola $y ^ { 2 } = 16 x$, which intersect the axis of the parabola at $A \& B$, respectively. If $C$ is the center of the circle through the points $P , A \& B$ and $\angle C P B = \theta$, then a value of $\tan \theta$ is:
(1) $\frac { 4 } { 3 }$
(2) $\frac { 1 } { 2 }$
(3) 2
(4) 3

jee-main 2018 Q70 Tangent Lines and Tangent Lengths View

Two parabolas with a common vertex and with axes along $x$-axis and $y$-axis, respectively, intersect each other in the first quadrant. if the length of the latus rectum of each parabola is 3 , then the equation of the common tangent to the two parabolas is?
(1) $3 ( x + y ) + 4 = 0$
(2) $8 ( 2 x + y ) + 3 = 0$
(3) $4 ( x + y ) + 3 = 0$
(4) $x + 2 y + 3 = 0$

jee-main 2018 Q71 Circle Identification and Classification View

Two sets $A$ and $B$ are as under: $A = \{ ( a , b ) \in R \times R : | a - 5 | < 1$ and $| b - 5 | < 1 \}$; $B = \left\{ ( a , b ) \in R \times R : 4 ( a - 6 ) ^ { 2 } + 9 ( b - 5 ) ^ { 2 } \leq 36 \right\}$. Then :
(1) neither $A \subset B$ nor $B \subset A$
(2) $B \subset A$
(3) $A \subset B$
(4) $A \cap B = \phi$ (an empty set)

jee-main 2018 Q71 Circle-Related Locus Problems View

If the tangent drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinates axes at the distinct points $A$ and $B$, then the locus of the midpoint of $AB$ is :
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$

jee-main 2018 Q72 Circle-Related Locus Problems View

If the tangents drawn to the hyperbola $4 y ^ { 2 } = x ^ { 2 } + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $A B$ is
(1) $x ^ { 2 } - 4 y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $4 x ^ { 2 } - y ^ { 2 } + 16 x ^ { 2 } y ^ { 2 } = 0$
(3) $4 x ^ { 2 } - y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(4) $x ^ { 2 } - 4 y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$

jee-main 2018 Q81 Intersection of Circles or Circle with Conic View

If the curves $y ^ { 2 } = 6 x , 9 x ^ { 2 } + b y ^ { 2 } = 16$ intersect each other at right angles, then the value of $b$ is:
(1) $\frac { 9 } { 2 }$
(2) 6
(3) $\frac { 7 } { 2 }$
(4) 4

jee-main 2019 Q68 Area and Geometric Measurement Involving Circles View

The tangent and the normal lines at the point $( \sqrt { 3 } , 1 )$ to the circle $x ^ { 2 } + y ^ { 2 } = 4$ and the $x$-axis form a triangle. The area of this triangle (in square units) is:
(1) $\frac { 1 } { 3 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { 4 } { \sqrt { 3 } }$
(4) $\frac { 1 } { \sqrt { 3 } }$

jee-main 2019 Q69 Circle-Related Locus Problems View

If a circle of radius $R$ passes through the origin $O$ and intersects the coordinate axes at $A$ and $B$, then the locus of the foot of perpendicular from $O$ on $AB$ is :
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ( x + y ) = R ^ { 2 } x y$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = 4 R ^ { 2 } x ^ { 2 } y ^ { 2 }$
(3) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 4 R ^ { 2 } x ^ { 2 } y ^ { 2 }$
(4) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 4 R x ^ { 2 } y ^ { 2 }$

jee-main 2019 Q69 Intersection of Circles or Circle with Conic View

The tangent to the parabola $y ^ { 2 } = 4 x$ at the point where it intersects the circle $x ^ { 2 } + y ^ { 2 } = 5$ in the first quadrant, passes through the point:
(1) $\left( \frac { 1 } { 4 } , \frac { 3 } { 4 } \right)$
(2) $\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)$
(3) $\left( - \frac { 1 } { 4 } , \frac { 1 } { 2 } \right)$
(4) $\left( \frac { 3 } { 4 } , \frac { 7 } { 4 } \right)$

jee-main 2019 Q69 Chord Length and Chord Properties View

The sum of the squares of the lengths of the chords intercepted on the circle, $x^2 + y^2 = 16$, by the lines, $x + y = n$, $n \in N$, where $N$ is the set of all natural numbers is:
(1) 210
(2) 105
(3) 320
(4) 160

jee-main 2019 Q69 Circle-Related Locus Problems View

If a tangent to the circle $x ^ { 2 } + y ^ { 2 } = 1$ intersects the coordinate axes at distinct points $P$ and $Q$, then the locus of the mid-point of $PQ$ is:
(1) $x ^ { 2 } + y ^ { 2 } - 16 x ^ { 2 } y ^ { 2 } = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 4 x ^ { 2 } y ^ { 2 } = 0$
(3) $x ^ { 2 } + y ^ { 2 } - 2 x y = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 2 x ^ { 2 } y ^ { 2 } = 0$

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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