LFM Pure

jee-main 2022 Q83 Distance from Center to Line View

Let a circle $C$ of radius 5 lie below the $x$-axis. The line $L _ { 1 } : 4 x + 3 y + 2 = 0$ passes through the centre $P$ of the circle $C$ and intersects the line $L _ { 2 } : 3 x - 4 y - 11 = 0$ at $Q$. The line $L _ { 2 }$ touches $C$ at the point $Q$. Then the distance of $P$ from the line $5 x - 12 y + 51 = 0$ is

jee-main 2022 Q83 Circle Equation Derivation View

If the length of the latus rectum of the ellipse $x ^ { 2 } + 4 y ^ { 2 } + 2 x + 8 y - \lambda = 0$ is 4 , and $l$ is the length of its major axis, then $\lambda + l$ is equal to $\_\_\_\_$ .

jee-main 2022 Q84 Chord Length and Chord Properties View

Let a circle $C : ( x - h ) ^ { 2 } + ( y - k ) ^ { 2 } = r ^ { 2 } , k > 0$, touch the $x$-axis at $( 1,0 )$. If the line $x + y = 0$ intersects the circle $C$ at $P$ and $Q$ such that the length of the chord $P Q$ is 2, then the value of $h + k + r$ is equal to $\_\_\_\_$.

jee-main 2022 Q85 Circle Equation Derivation View

Let $P _ { 1 }$ be a parabola with vertex $( 3,2 )$ and focus $( 4,4 )$ and $P _ { 2 }$ be its mirror image with respect to the line $x + 2 y = 6$. Then the directrix of $P _ { 2 }$ is $x + 2 y =$ $\_\_\_\_$.

jee-main 2022 Q85 Circles Tangent to Each Other or to Axes View

The sum of diameters of the circles that touch (i) the parabola $75x ^ { 2 } = 64(5y - 3)$ at the point $\left(\frac { 8 } { 5 }, \frac { 6 } { 5 }\right)$ and (ii) the $y$-axis, is equal to $\_\_\_\_$.

jee-main 2022 Q86 Circle Equation Derivation View

Let the equation of two diameters of a circle $x ^ { 2 } + y ^ { 2 } - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0,\infty)$ of the tangent to the hyperbola $3x ^ { 2 } - y ^ { 2 } = 3$ passing through the centre of the circle is equal to $\_\_\_\_$.

jee-main 2022 Q87 Tangent Lines and Tangent Lengths View

Two tangent lines $l _ { 1 }$ and $l _ { 2 }$ are drawn from the point $( 2,0 )$ to the parabola $2 y ^ { 2 } = - x$. If the lines $l _ { 1 }$ and $l _ { 2 }$ are also tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r$, then $17 r ^ { 2 }$ is equal to $\_\_\_\_$.

jee-main 2022 Q88 Tangent Lines and Tangent Lengths View

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 4 } = 1$ meet at the point $R ( \sqrt { 2 } , 2 \sqrt { 2 } - 2 )$. If $S$ is the focus of the ellipse on its negative major axis, then $S P ^ { 2 } + S Q ^ { 2 }$ is equal to $\_\_\_\_$.

jee-main 2023 Q64 Area and Geometric Measurement Involving Circles View

Let a circle $C_1$ be obtained on rolling the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ upwards 4 units on the tangent $T$ to it at the point $(3,2)$. Let $C_2$ be the image of $C_1$ in $T$. Let $A$ and $B$ be the centers of circles $C_1$ and $C_2$ respectively, and $M$ and $N$ be respectively the feet of perpendiculars drawn from $A$ and $B$ on the $x$-axis. Then the area of the trapezium AMNB is:
(1) $22 + \sqrt{2}$
(2) $41 + \sqrt{2}$
(3) $3 + 2\sqrt{2}$
(4) $21 + \sqrt{2}$

jee-main 2023 Q65 Circle-Related Locus Problems View

A line segment $AB$ of length $\lambda$ moves such that the points $A$ and $B$ remain on the periphery of a circle of radius $\lambda$. Then the locus of the point, that divides the line segment $AB$ in the ratio $2:3$, is a circle of radius
(1) $\frac{3}{5}\lambda$
(2) $\frac{2}{3}\lambda$
(3) $\frac{\sqrt{19}}{5}\lambda$
(4) $\frac{\sqrt{19}}{7}\lambda$

jee-main 2023 Q66 Area and Geometric Measurement Involving Circles View

Let the ellipse $E$: $x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A$, $P$ and the origin $O$ is $\frac{m}{n}$, where $m$ and $n$ are coprime, then $m - n$ is equal to
(1) 16
(2) 15
(3) 17
(4) 18

jee-main 2023 Q67 Circle-Related Locus Problems View

Let $A$ be the point $(1, 2)$ and $B$ be any point on the curve $x ^ { 2 } + y ^ { 2 } = 16$. If the centre of the locus of the point $P$, which divides the line segment $AB$ in the ratio $3 : 2$ is the point $C( \alpha , \beta )$, then the length of the line segment $AC$ is
(1) $\frac { 3 \sqrt { 5 } } { 5 }$
(2) $\frac { 4 \sqrt { 5 } } { 5 }$
(3) $\frac { 2 \sqrt { 5 } } { 5 }$
(4) $\frac { 6 \sqrt { 5 } } { 5 }$

jee-main 2023 Q67 Circle Equation Derivation View

Let the centre of a circle $C$ be $( \alpha , \beta )$ and its radius $r < 8$. Let $3 x + 4 y = 24$ and $3 x - 4 y = 32$ be two tangents and $4 x + 3 y = 1$ be a normal to $C$. Then $( \alpha - \beta + r )$ is equal to
(1) 7
(2) 5
(3) 6
(4) 9

jee-main 2023 Q67 Tangent Lines and Tangent Lengths View

The number of common tangents, to the circles $x ^ { 2 } + y ^ { 2 } - 18 x - 15 y + 131 = 0$ and $x ^ { 2 } + y ^ { 2 } - 6 x - 6 y - 7 = 0$, is
(1) 3
(2) 1
(3) 4
(4) 2

jee-main 2023 Q68 Chord Length and Chord Properties View

The set of all values of $a^2$ for which the line $x + y = 0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2}, \frac{1-a}{2}\right)$ on the circle $2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$, is equal to:
(1) $(8, \infty)$
(2) $(0, 4]$
(3) $(4, \infty)$
(4) $(2, 12]$

jee-main 2023 Q68 Circle Equation Derivation View

The points of intersection of the line $a x + b y = 0 , ( \mathrm { a } \neq \mathrm { b } )$ and the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } - 2 \mathrm { x } = 0$ are $A ( \alpha , 0 )$ and $B ( 1 , \beta )$. The image of the circle with $A B$ as a diameter in the line $\mathrm { x } + \mathrm { y } + 2 = 0$ is:
(1) $x ^ { 2 } + y ^ { 2 } + 5 x + 5 y + 12 = 0$
(2) $x ^ { 2 } + y ^ { 2 } + 3 x + 5 y + 8 = 0$
(3) $x ^ { 2 } + y ^ { 2 } + 3 x + 3 y + 4 = 0$
(4) $x ^ { 2 } + y ^ { 2 } - 5 x - 5 y + 12 = 0$

jee-main 2023 Q68 Area and Geometric Measurement Involving Circles View

Let $P(a_{1}, b_{1})$ and $Q(a_{2}, b_{2})$ be two distinct points on a circle with center $C(\sqrt{2}, \sqrt{3})$. Let $O$ be the origin and $OC$ be perpendicular to both $CP$ and $CQ$. If the area of the triangle $OCP$ is $\frac{\sqrt{35}}{2}$, then $a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2}$ is equal to $\_\_\_\_$

jee-main 2023 Q69 Circle-Related Locus Problems View

The locus of the middle points of the chords of the circle $C _ { 1 } : ( x - 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 4$ which subtend an angle $\theta _ { i }$ at the centre of the circle $C _ { i }$, is a circle of radius $r _ { i }$. If $\theta _ { 1 } = \frac { \pi } { 3 } , \theta _ { 3 } = \frac { 2 \pi } { 3 }$ and $r _ { 1 } ^ { 2 } = r _ { 2 } ^ { 2 } + r _ { 3 } ^ { 2 }$, then $\theta _ { 2 }$ is equal to
(1) $\frac { \pi } { 4 }$
(2) $\frac { 3 \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 2 }$

jee-main 2023 Q70 Area and Geometric Measurement Involving Circles View

If the tangents at the points $P$ and $Q$ on the circle $x^{2} + y^{2} - 2x + y = 5$ meet at the point $R\left(\frac{9}{4}, 2\right)$, then the area of the triangle $PQR$ is
(1) $\frac{5}{4}$
(2) $\frac{13}{8}$
(3) $\frac{5}{8}$
(4) $\frac{13}{4}$

jee-main 2023 Q70 Inscribed/Circumscribed Circle Computations View

Let $O$ be the origin and $O P$ and $O Q$ be the tangents to the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + 8 = 0$ at the points $P$ and $Q$ on it. If the circumcircle of the triangle $O P Q$ passes through the point $\left( \alpha , \frac { 1 } { 2 } \right)$, then a value of $\alpha$ is
(1) $\frac { 3 } { 2 }$
(2) $- \frac { 1 } { 2 }$
(3) $\frac { 5 } { 2 }$
(4) 1

jee-main 2023 Q70 Tangent Lines and Tangent Lengths View

A circle with centre $( 2,3 )$ and radius 4 intersects the line $x + y = 3$ at the points $P$ and $Q$. If the tangents at $P$ and $Q$ intersect at the point $S ( \alpha , \beta )$, then $4 \alpha - 7 \beta$ is equal to $\_\_\_\_$

jee-main 2023 Q70 Tangent Lines and Tangent Lengths View

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR)^{2}$ is equal to
(1) 64
(2) 76
(3) 81
(4) 72

jee-main 2023 Q70 Circle Equation Derivation View

Consider a circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 4 x - 2 y = \alpha - 5$. Let its mirror image in the line $y = 2 x + 1$ be another circle $C _ { 2 } : 5 x ^ { 2 } + 5 y ^ { 2 } - 10 f x - 10 g y + 36 = 0$. Let $r$ be the radius of $C _ { 2 }$. Then $\alpha + r$ is equal to $\_\_\_\_$

jee-main 2023 Q70 Circles Tangent to Each Other or to Axes View

Two circles in the first quadrant of radii $r _ { 1 }$ and $r _ { 2 }$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $x + y = 2$. Then $r _ { 1 } { } ^ { 2 } + r _ { 2 } { } ^ { 2 } - r _ { 1 } r _ { 2 }$ is equal to $\_\_\_\_$ .

jee-main 2023 Q71 Tangent Lines and Tangent Lengths View

Points $P ( - 3,2 ) , Q ( 9,10 )$ and $R ( \alpha , 4 )$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x - k y = 1$, then $k$ is equal to $\_\_\_\_$.

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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