LFM Pure

jee-main 2021 Q64 Circle-Line Intersection and Point Conditions View

Let the circle $S : 36 x ^ { 2 } + 36 y ^ { 2 } - 108 x + 120 y + C = 0$ be such that it neither intersects nor touches the coordinate axes. If the point of intersection of the lines, $x - 2 y = 4$ and $2 x - y = 5$ lies inside the circle $S$, then:
(1) $\frac { 25 } { 9 } < C < \frac { 13 } { 3 }$
(2) $100 < C < 165$
(3) $81 < C < 156$
(4) $100 < C < 156$

jee-main 2021 Q64 Optimization on a Circle View

Let $r _ { 1 }$ and $r _ { 2 }$ be the radii of the largest and smallest circles, respectively, which pass through the point $( - 4,1 )$ and having their centres on the circumference of the circle $x ^ { 2 } + y ^ { 2 } + 2 x + 4 y - 4 = 0$. If $\frac { r _ { 1 } } { r _ { 2 } } = a + b \sqrt { 2 }$, then $a + b$ is equal to:
(1) 3
(2) 11
(3) 5
(4) 7

jee-main 2021 Q64 Circle-Related Locus Problems View

If two tangents drawn from a point $P$ to the parabola $y ^ { 2 } = 16 ( x - 3 )$ are at right angles, then the locus of point $P$ is: (1) $x + 4 = 0$ (2) $x + 2 = 0$ (3) $x + 3 = 0$ (4) $x + 1 = 0$

jee-main 2021 Q65 Optimization on a Circle View

Let $A ( 1,4 )$ and $B ( 1 , - 5 )$ be two points. Let $P$ be a point on the circle $( ( x - 1 ) ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$, such that $( P A ) ^ { 2 } + ( P B ) ^ { 2 }$ have maximum value, then the points, $P , A$ and $B$ lie on
(1) a hyperbola
(2) a straight line
(3) an ellipse
(4) a parabola

jee-main 2021 Q65 Inscribed/Circumscribed Circle Computations View

In a triangle $PQR$, the co-ordinates of the points $P$ and $Q$ are $(-2, 4)$ and $(4, -2)$ respectively. If the equation of the perpendicular bisector of $PR$ is $2x - y + 2 = 0$, then the centre of the circumcircle of the $\triangle PQR$ is:
(1) $(-1, 0)$
(2) $(-2, -2)$
(3) $(0, 2)$
(4) $(1, 4)$

jee-main 2021 Q65 Circle-Related Locus Problems View

Let $S _ { 1 } : x ^ { 2 } + y ^ { 2 } = 9$ and $S _ { 2 } : ( x - 2 ) ^ { 2 } + y ^ { 2 } = 1$.
Then the locus of center of a variable circle $S$ which touches $S _ { 1 }$ internally and $S _ { 2 }$ externally always passes through the points:
(1) $( 0 , \pm \sqrt { 3 } )$
(2) $\left( \frac { 1 } { 2 } , \pm \frac { \sqrt { 5 } } { 2 } \right)$
(3) $\left( 2 , \pm \frac { 3 } { 2 } \right)$
(4) $( 1 , \pm 2 )$

jee-main 2021 Q65 Area and Geometric Measurement Involving Circles View

If the curve $x ^ { 2 } + 2 y ^ { 2 } = 2$ intersects the line $x + y = 1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is
(1) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(2) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)$
(3) $\frac { \pi } { 2 } + \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$
(4) $\frac { \pi } { 2 } - \tan ^ { - 1 } \left( \frac { 1 } { 4 } \right)$

jee-main 2021 Q65 Area and Geometric Measurement Involving Circles View

Two tangents are drawn from a point $P$ to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$, such that the angle between these tangents is $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right)$, where $\tan ^ { - 1 } \left( \frac { 12 } { 5 } \right) \in ( 0 , \pi )$. If the centre of the circle is denoted by $C$ and these tangents touch the circle at points $A$ and $B$, then the ratio of the areas of $\triangle P A B$ and $\triangle C A B$ is:
(1) $11 : 4$
(2) $9 : 4$
(3) $3 : 1$
(4) $2 : 1$

jee-main 2021 Q65 Circle Equation Derivation View

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $x$-axis at a distance $R$ and $S ( > \mathrm { R } )$ respectively from the origin, is :
(1) $2 ( S - R )$
(2) $2 ( S + R )$
(3) $4 ( S - R )$
(4) $4 ( S + R )$

jee-main 2021 Q66 Circle Equation Derivation View

The line $2x - y + 1 = 0$ is a tangent to the circle at the point $(2, 5)$ and the centre of the circle lies on $x - 2y = 4$. Then, the radius of the circle is:
(1) $3\sqrt{5}$
(2) $5\sqrt{3}$
(3) $5\sqrt{4}$
(4) $4\sqrt{5}$

jee-main 2021 Q66 Circle Equation Derivation View

The image of the point $( 3,5 )$ in the line $x - y + 1 = 0$, lies on :
(1) $( x - 2 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 4$
(2) $( x - 4 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 8$
(3) $( x - 4 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
(4) $( x - 2 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 12$

jee-main 2021 Q66 Circle Equation Derivation View

If the three normals drawn to the parabola, $y ^ { 2 } = 2 x$ pass through the point $( a , 0 ) , a \neq 0$, then $a$ must be greater than :
(1) $\frac { 1 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) - 1
(4) 1

jee-main 2021 Q66 Inscribed/Circumscribed Circle Computations View

Let the tangent to the circle $x ^ { 2 } + y ^ { 2 } = 25$ at the point $R ( 3,4 )$ meet $x$-axis and $y$-axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $O P Q$, then $r ^ { 2 }$ is equal to
(1) $\frac { 529 } { 64 }$
(2) $\frac { 125 } { 72 }$
(3) $\frac { 625 } { 72 }$
(4) $\frac { 585 } { 66 }$

jee-main 2021 Q66 Tangent Lines and Tangent Lengths View

The line $12 x \cos \theta + 5 y \sin \theta = 60$ is tangent to which of the following curves ?
(1) $x ^ { 2 } + y ^ { 2 } = 30$
(2) $144 x ^ { 2 } + 25 y ^ { 2 } = 3600$
(3) $x ^ { 2 } + y ^ { 2 } = 169$
(4) $25 x ^ { 2 } + 12 y ^ { 2 } = 3600$

jee-main 2021 Q67 Intersection of Circles or Circle with Conic View

Choose the incorrect statement about the two circles whose equations are given below: $x ^ { 2 } + y ^ { 2 } - 10 x - 10 y + 41 = 0$ and $x ^ { 2 } + y ^ { 2 } - 16 x - 10 y + 80 = 0$
(1) Distance between two centres is the average of radii of both the circles.
(2) Both circles' centres lie inside region of one another.
(3) Both circles pass through the centre of each other.
(4) Circles have two intersection points.

jee-main 2021 Q67 Intersection of Circles or Circle with Conic View

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 } = 1$ and the circle $x^2 + y^2 = 3$ at their points of intersection. Then $\tan\theta$ is equal to:
(1) $\frac{4}{\sqrt{3}}$
(2) $\frac{2}{\sqrt{3}}$
(3) $2$
(4) $\frac{5}{2\sqrt{3}}$

jee-main 2021 Q67 Tangent Lines and Tangent Lengths View

A tangent is drawn to the parabola $y ^ { 2 } = 6 x$ which is perpendicular to the line $2 x + y = 1$. Which of the following points does NOT lie on it?
(1) $( 0,3 )$
(2) $( 4,5 )$
(3) $( 5,4 )$
(4) $( - 6,0 )$

jee-main 2021 Q67 Circle-Related Locus Problems View

The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is :
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 16 x ^ { 2 } + 9 y ^ { 2 } = 0$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 144 y ^ { 2 } = 0$
(3) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } - 16 y ^ { 2 } = 0$
(4) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } - 9 x ^ { 2 } + 16 y ^ { 2 } = 0$

jee-main 2021 Q67 Tangent Lines and Tangent Lengths View

Let $L$ be a tangent line to the parabola $y ^ { 2 } = 4 x - 20$ at (6, 2). If $L$ is also a tangent to the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { b } = 1$, then the value of $b$ is equal to:
(1) 11
(2) 14
(3) 16
(4) 20

jee-main 2021 Q68 Intersection of Circles or Circle with Conic View

If the curves, $\frac { x ^ { 2 } } { a } + \frac { y ^ { 2 } } { b } = 1$ and $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ intersect each other at an angle of $90 ^ { \circ }$, then which of the following relations is TRUE?
(1) $a - c = b + d$
(2) $a - b = c - d$
(3) $a + b = c + d$
(4) $a b = \frac { c + d } { a + b }$

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 Inscribed/Circumscribed Circle Computations View

Consider a triangle having vertices $A ( - 2,3 ) , B ( 1,9 )$ and $C ( 3,8 )$. If a line $L$ passing through the circumcentre of triangle $ABC$, bisects line $BC$, and intersects $y$-axis at point $\left( 0 , \frac { \alpha } { 2 } \right)$, then the value of real number $\alpha$ is $\underline{\hspace{1cm}}$.

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

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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