LFM Pure

jee-main 2016 Q76 Optimization on Conics View

If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} = 1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in sq. units) of the triangle $OAB$ is:
(1) $\frac{9}{2}$
(2) $9$
(3) $9\sqrt{3}$
(4) $\frac{\sqrt{3}}{2}$

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 Optimization on Conics View

Let $P$ be a point on the parabola $x ^ { 2 } = 4 y$. If the distance of $P$ from the center of the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 8 = 0$ is minimum, then the equation of the tangent to the parabola at $P$ is
(1) $x + y + 1 = 0$
(2) $x + 4 y - 2 = 0$
(3) $x + 2 y = 0$
(4) $x - y + 3 = 0$

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 2019 Q62 Circle Equation and Properties via Complex Number Manipulation View

All the points in the set $S = \left\{ \frac { \alpha + i } { \alpha - i } , \alpha \in R \right\} , i = \sqrt { - 1 }$ lie on a
(1) straight line whose slope is - 1
(2) circle whose radius is $\sqrt { 2 }$
(3) circle whose radius is 1
(4) straight line whose slope is 1

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

jee-main 2019 Q69 Area and Geometric Measurement Involving Circles View

A rectangle is inscribed in a circle with a diameter lying along the line $3 y = x + 7$. If the two adjacent vertices of the rectangle are $( - 8,5 )$ and $( 6,5 )$, then the area of the rectangle (in sq. units) is:
(1) 72
(2) 98
(3) 56
(4) 84

jee-main 2019 Q70 Circles Tangent to Each Other or to Axes View

If a circle $C$ passing through the point $( 4,0 )$ touches the circle $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12$ externally at the point $( 1 , - 1 )$, then the radius of $C$ is:
(1) 4 units
(2) 5 units
(3) $2 \sqrt { 5 }$ units
(4) $\sqrt { 57 }$ units

jee-main 2019 Q70 Tangent Lines and Tangent Lengths View

The equation of a tangent to the parabola, $x ^ { 2 } = 8 y$, which makes an angle $\theta$ with the positive direction of $x$-axis, is
(1) $y = x \tan \theta + 2 \cot \theta$
(2) $y = x \tan \theta - 2 \cot \theta$
(3) $x = y \cot \theta + 2 \tan \theta$
(4) $x = y \cot \theta - 2 \tan \theta$

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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