LFM Pure

jee-main 2011 Q68 Circles Tangent to Each Other or to Axes View

The two circles $x^{2}+y^{2}=ax$ and $x^{2}+y^{2}=c^{2}$ $(c>0)$ touch each other if
(1) $|a|=c$
(2) $a=2c$
(3) $|a|=2c$
(4) $2|a|=c$

jee-main 2012 Q69 Circle Equation Derivation View

The equation of the circle passing through the point $( 1,2 )$ and through the points of intersection of $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 21 = 0$ and $3 x + 4 y + 5 = 0$ is given by
(1) $x ^ { 2 } + y ^ { 2 } + 2 x + 2 y + 11 = 0$
(2) $x ^ { 2 } + y ^ { 2 } - 2 x + 2 y - 7 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 2 x - 2 y - 3 = 0$
(4) $x ^ { 2 } + y ^ { 2 } + 2 x + 2 y - 11 = 0$

jee-main 2012 Q69 Chord Length and Chord Properties View

If the line $y = m x + 1$ meets the circle $x ^ { 2 } + y ^ { 2 } + 3 x = 0$ in two points equidistant from and on opposite sides of $x$-axis, then
(1) $3 m + 2 = 0$
(2) $3 m - 2 = 0$
(3) $2 m + 3 = 0$
(4) $2 m - 3 = 0$

jee-main 2012 Q70 Tangent Lines and Tangent Lengths View

The number of common tangents of the circles given by $x ^ { 2 } + y ^ { 2 } - 8 x - 2 y + 1 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 8 y = 0$ is
(1) one
(2) four
(3) two
(4) three

jee-main 2012 Q70 Circle Equation Derivation View

The equation of the circle passing through the point $(1,0)$ and $(0,1)$ and having the smallest radius is
(1) $x^{2}+y^{2}-2x-2y+1=0$
(2) $x^{2}+y^{2}+2x+2y-7=0$
(3) $x^{2}+y^{2}-x-y=0$
(4) $x^{2}+y^{2}+x+y-2=0$

jee-main 2012 Q78 Circle Equation Derivation View

The length of the diameter of the circle which touches the $x$-axis at the point $(1,0)$ and passes through the point $(2,3)$ is
(1) $\frac{10}{3}$
(2) $\frac{3}{5}$
(3) $\frac{6}{5}$
(4) $\frac{5}{3}$

jee-main 2013 Q69 Circles Tangent to Each Other or to Axes View

If the circle $x ^ { 2 } + y ^ { 2 } - 6 x - 8 y + \left( 25 - a ^ { 2 } \right) = 0$ touches the axis of $x$, then $a$ equals.
(1) 0
(2) $\pm 4$
(3) $\pm 2$
(4) $\pm 3$

jee-main 2013 Q70 Circle Equation Derivation View

If each of the lines $5 x + 8 y = 13$ and $4 x - y = 3$ contains a diameter of the circle $x ^ { 2 } + y ^ { 2 } - 2 \left( a ^ { 2 } - 7 a + 11 \right) x - 2 \left( a ^ { 2 } - 6 a + 6 \right) y + b ^ { 3 } + 1 = 0$, then :
(1) $a = 5$ and $b \notin ( - 1,1 )$
(2) $a = 1$ and $b \notin ( - 1,1 )$
(3) $a = 2$ and $b \notin ( - \infty , 1 )$
(4) $a = 5$ and $b \in ( - \infty , 1 )$

jee-main 2013 Q70 Circle Equation Derivation View

Statement 1: The only circle having radius $\sqrt { 10 }$ and a diameter along line $2x + y = 5$ is $x ^ { 2 } + y ^ { 2 } - 6x + 2y = 0$. Statement 2: $2x + y = 5$ is a normal to the circle $x ^ { 2 } + y ^ { 2 } - 6x + 2y = 0$.
(1) Statement 1 is false; Statement 2 is true.
(2) Statement 1 is true; Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(3) Statement 1 is true; Statement 2 is false.
(4) Statement 1 is true; Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

jee-main 2013 Q71 Circle Equation Derivation View

The circle passing through $(1, -2)$ and touching the axis of $x$ at $(3, 0)$ also passes through the point
(1) $(5, -2)$
(2) $(-2, 5)$
(3) $(-5, 2)$
(4) $(2, -5)$

jee-main 2013 Q72 Tangent Lines and Tangent Lengths View

Given: A circle, $2x^2 + 2y^2 = 5$ and a parabola, $y^2 = 4\sqrt{5}x$. Statement-I: An equation of a common tangent to these curves is $y = x + \sqrt{5}$. Statement-II: If the line, $y = mx + \frac{\sqrt{5}}{m}$ $(m \neq 0)$ is their common tangent, then $m$ satisfies $m^4 - 3m^2 + 2 = 0$.
(1) Statement-I is true; Statement-II is false.
(2) Statement-I is false; Statement-II is true.
(3) Statement-I is true; Statement-II is true; Statement-II is a correct explanation for Statement-I.
(4) Statement-I is true; Statement-II is true; Statement-II is not a correct explanation for Statement-I.

jee-main 2013 Q73 Circle Equation Derivation View

The equation of the circle passing through the foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$, and having centre at $(0, 3)$ is
(1) $x^2 + y^2 - 6y - 5 = 0$
(2) $x^2 + y^2 - 6y + 5 = 0$
(3) $x^2 + y^2 - 6y - 7 = 0$
(4) $x^2 + y^2 - 6y + 7 = 0$

jee-main 2014 Q69 Circle Equation Derivation View

The equation of the circle described on the chord $3 x + y + 5 = 0$ of the circle $x ^ { 2 } + y ^ { 2 } = 16$ as the diameter is
(1) $x ^ { 2 } + y ^ { 2 } + 3 x + y + 1 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 22 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 11 = 0$
(4) $x ^ { 2 } + y ^ { 2 } + 3 x + y - 2 = 0$

jee-main 2014 Q70 Circles Tangent to Each Other or to Axes View

Let $C$ be the circle with center at $( 1,1 )$ and radius $= 1$. If $T$ is the circle centered at $( 0 , y )$, passing through the origin and touching the circle $C$ externally, then the radius of $T$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \sqrt { 3 } } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 2 }$

jee-main 2014 Q71 Circle-Related Locus Problems View

The locus of the foot of perpendicular drawn from the centre of the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 6$ on any tangent to it is
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$
(3) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(4) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$

jee-main 2014 Q71 Circle-Related Locus Problems View

Let $a$ and $b$ be any two numbers satisfying $\frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } = \frac { 1 } { 4 }$. Then, the foot of perpendicular from the origin on the variable line $\frac { x } { a } + \frac { y } { b } = 1$ lies on:
(1) A circle of radius $= 2$
(2) A hyperbola with each semi-axis $= \sqrt { 2 }$.
(3) A hyperbola with each semi-axis $= 2$
(4) A circle of radius $= \sqrt { 2 }$

jee-main 2014 Q72 Circle-Line Intersection and Point Conditions View

If the point $( 1,4 )$ lies inside the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval
(1) $( 25,39 )$
(2) $( 25,29 )$
(3) $( 0,25 )$
(4) $( 9,25 )$

jee-main 2015 Q66 Tangent Lines and Tangent Lengths View

The number of common tangents to the circles $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 + 6x + 18y + 26 = 0$, is:
(1) 1
(2) 2
(3) 3
(4) 4

jee-main 2015 Q70 Tangent Lines and Tangent Lengths View

The number of common tangents to the circles $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0$ and $x ^ { 2 } + y ^ { 2 } + 6 x + 18 y + 26 = 0$, is
(1) 4
(2) 1
(3) 2
(4) 3

jee-main 2015 Q71 Circle-Related Locus Problems View

Let $O$ be the vertex and $Q$ be any point on the parabola, $x ^ { 2 } = 8 y$. If the point $P$ divides the line segment $OQ$ internally in the ratio $1 : 3$, then the locus of $P$ is
(1) $x ^ { 2 } = 2 y$
(2) $x ^ { 2 } = y$
(3) $y ^ { 2 } = x$
(4) $y ^ { 2 } = 2 x$

jee-main 2015 Q72 Area and Geometric Measurement Involving Circles View

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$, is
(1) 27
(2) $\frac { 27 } { 4 }$
(3) 18
(4) $\frac { 27 } { 2 }$

jee-main 2015 Q86 Circle-Related Locus Problems View

Locus of the image of the point $(2, 3)$ in the line $(2x - 3y + 4) + k(x - 2y + 3) = 0$, $k \in \mathbb{R}$, is a:
(1) straight line parallel to $x$-axis
(2) straight line parallel to $y$-axis
(3) circle of radius $\sqrt{2}$
(4) circle of radius $\sqrt{3}$

jee-main 2016 Q70 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 2016 Q71 Tangent Lines and Tangent Lengths View

Equation of the tangent to the circle, at the point $( 1 , - 1 )$, whose center, is the point of intersection of the straight lines $x - y = 1$ and $2 x + y = 3$ is:
(1) $x + 4 y + 3 = 0$
(2) $3 x - y - 4 = 0$
(3) $x - 3 y - 4 = 0$
(4) $4 x + y - 3 = 0$

jee-main 2016 Q72 Optimization on a Circle View

$P$ and $Q$ are two distinct points on the parabola, $y ^ { 2 } = 4 x$, with parameters $t$ and $t _ { 1 }$, respectively. If the normal at $P$ passes through $Q$, then the minimum value of $t _ { 1 } ^ { 2 }$, is
(1) 8
(2) 4
(3) 6
(4) 2

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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