LFM Pure

isi-entrance 2017 Q2 Chord Length and Chord Properties View

Consider a circle of radius 6 as given in the diagram below. Let $B$, $C , D$ and $E$ be points on the circle such that $B D$ and $C E$, when extended, intersect at $A$. If $A D$ and $A E$ have length 5 and 4 respectively, and $D B C$ is a right angle, then show that the length of $B C$ is $\frac { 12 + 9 \sqrt { 15 } } { 5 }$.

isi-entrance 2017 Q12 Circle-Related Locus Problems View

Let $C$ be a circle of area $A$ with centre at $O$. Let $P$ be a moving point such that its distance from $O$ is always twice the length of a tangent drawn from $P$ to the circle. Then the point $P$ must move along
(A) the sides of a square centred at $O$, with area $\frac{4}{3}A$.
(B) the sides of an equilateral triangle centred at $O$, with area $4A$.
(C) a circle centred at $O$, with area $\frac{4}{3}A$.
(D) a circle centred at $O$, with area $4A$.

isi-entrance 2017 Q24 Inscribed/Circumscribed Circle Computations View

Let $P = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $Q = \left(-\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}}\right)$ be two vertices of a regular polygon having 12 sides such that $PQ$ is a diameter of the circle circumscribing the polygon. Which of the following points is not a vertex of this polygon?
(A) $\left(\frac{\sqrt{3}-1}{2\sqrt{2}}, \frac{\sqrt{3}+1}{2\sqrt{2}}\right)$
(B) $\left(\frac{\sqrt{3}+1}{2\sqrt{2}}, \frac{\sqrt{3}-1}{2\sqrt{2}}\right)$
(C) $\left(\frac{\sqrt{3}+1}{2\sqrt{2}}, \frac{1-\sqrt{3}}{2\sqrt{2}}\right)$
(D) $\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.

isi-entrance 2018 Q2 Area and Geometric Measurement Involving Circles View

Suppose that $P Q$ and $R S$ are two chords of a circle intersecting at a point $O$. It is given that $P O = 3 \mathrm {~cm}$ and $S O = 4 \mathrm {~cm}$. Moreover, the area of the triangle $P O R$ is $7 \mathrm {~cm} ^ { 2 }$. Find the area of the triangle $Q O S$.

isi-entrance 2019 Q20 Inscribed/Circumscribed Circle Computations View

In the following picture, $A B C$ is an isosceles triangle with an inscribed circle with center $O$. Let $P$ be the mid-point of $B C$. If $A B = A C = 15$ and $B C = 10$, then $O P$ equals:
(A) $\frac { \sqrt { 5 } } { \sqrt { 2 } }$
(B) $\frac { 5 } { \sqrt { 2 } }$
(C) $2 \sqrt { 5 }$
(D) $5 \sqrt { 2 }$.

isi-entrance 2019 Q24 Inscribed/Circumscribed Circle Computations View

Chords $A B$ and $C D$ of a circle intersect at right angle at the point $P$. If the lengths of $A P , P B , C P , P D$ are $2,6,3,4$ units respectively, then the radius of the circle is:
(A) 4
(B) $\frac { \sqrt { 65 } } { 2 }$
(C) $\frac { \sqrt { 66 } } { 2 }$
(D) $\frac { \sqrt { 67 } } { 2 }$

isi-entrance 2020 Q11 Area and Geometric Measurement Involving Circles View

Two vertices of a square lie on a circle of radius $r$ and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is
(A) $\frac { 3 r } { 2 }$
(B) $\frac { 4 r } { 3 }$
(C) $\frac { 6 r } { 5 }$
(D) $\frac { 8 r } { 5 }$.

isi-entrance 2021 Q10 Area and Geometric Measurement Involving Circles View

Consider the curves $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0,9 x ^ { 2 } + 4 y ^ { 2 } - 900 = 0$ and $y ^ { 2 } - 6 y - 6 x + 51 = 0$. The maximum number of disjoint regions into which these curves divide the $XY$-plane (excluding the curves themselves), is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .

isi-entrance 2021 Q28 Optimization on a Circle View

If two real numbers $x$ and $y$ satisfy $( x + 5 ) ^ { 2 } + ( y - 10 ) ^ { 2 } = 196$, then the minimum possible value of $x ^ { 2 } + 2 x + y ^ { 2 } - 4 y$ is
(A) $271 - 112 \sqrt { 5 }$.
(B) $14 - 4 \sqrt { 5 }$.
(C) $276 - 112 \sqrt { 5 }$.
(D) $9 - 4 \sqrt { 5 }$.

isi-entrance 2023 Q6 Area and Geometric Measurement Involving Circles View

Consider a triangle with vertices $( 0,0 ) , ( 1,2 )$ and $( - 4,2 )$. Let $A$ be the area of the triangle and $B$ be the area of the circumcircle of the triangle. Then $\frac { B } { A }$ equals
(A) $\frac { \pi } { 2 }$.
(B) $\frac { 5 \pi } { 4 }$.
(C) $\frac { 3 } { \sqrt { 2 } } \pi$.
(D) $2 \pi$.

isi-entrance 2023 Q13 Intersection of Circles or Circle with Conic View

For real numbers $a , b , c , d , a ^ { \prime } , b ^ { \prime } , c ^ { \prime } , d ^ { \prime }$, consider the system of equations $$\begin{aligned} a x ^ { 2 } + a y ^ { 2 } + b x + c y + d & = 0 \\ a ^ { \prime } x ^ { 2 } + a ^ { \prime } y ^ { 2 } + b ^ { \prime } x + c ^ { \prime } y + d ^ { \prime } & = 0 \end{aligned}$$ If $S$ denotes the set of all real solutions $( x , y )$ of the above system of equations, then the number of elements in $S$ can never be
(A) 0.
(B) 1.
(C) 2.
(D) 3.

isi-entrance 2023 Q22 Area and Geometric Measurement Involving Circles View

In the following figure, $O A B$ is a quarter-circle. The unshaded region is a circle to which $O A$ and $C D$ are tangents. If $C D$ is of length 10 and is parallel to $O A$, then the area of the shaded region in the above figure equals
(A) $25 \pi$.
(B) $50 \pi$.
(C) $75 \pi$.
(D) $100 \pi$.

isi-entrance 2024 Q8 Chord Length and Chord Properties View

The angle subtended at the origin by the common chord of the circles $x^2 + y^2 - 6x - 6y = 0$ and $x^2 + y^2 = 36$ is
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi/3$
(D) $2\pi/3$

isi-entrance 2024 Q21 Circle-Line Intersection and Point Conditions View

Consider points of the form $\left(n, n^k\right)$, where $n$ and $k$ are integers with $n \geq 0$, $k \geq 1$. How many such points are strictly inside the circle of radius 10 with centre at the origin?
(A) 11
(B) 12
(C) 15
(D) 17

isi-entrance 2024 Q27 Area and Geometric Measurement Involving Circles View

In the adjoining figure, $C$ is the centre of the circle drawn, $A, F, E$ lie on the circle and $BCDF$ is a rectangle. If $\frac{DE}{AB} = 2$, then $\frac{FE}{FA}$ equals
(A) $\sqrt{\frac{3}{2}}$
(B) $\sqrt{2}$
(C) $\sqrt{\frac{5}{2}}$
(D) $\sqrt{3}$

isi-entrance 2026 Q7 Chord Length and Chord Properties View

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(a) $36 ^ { \circ }$.
(B) $29 ^ { \circ }$.
(C) $7 ^ { \circ }$.
(D) $25 ^ { \circ }$.

isi-entrance 2026 Q17 Circle-Related Locus Problems View

A triangle $A B C$ has a fixed base $B C$. If $A B : A C = 1 : 2$, then the locus of the vertex $A$ is
(a) a circle whose centre is the midpoint of $B C$.
(b) a circle whose centre is on the line $B C$ but not the midpoint of $B C$.
(c) a straight line.
(d) none of the above.

jee-advanced 2008 Q1 Intersection of Circles or Circle with Conic View

Consider the two curves $$\begin{aligned} & C _ { 1 } : y ^ { 2 } = 4 x \\ & C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 6 x + 1 = 0 \end{aligned}$$ Then,
(A) $C _ { 1 }$ and $C _ { 2 }$ touch each other only at one point
(B) $C _ { 1 }$ and $C _ { 2 }$ touch each other exactly at two points
(C) $C _ { 1 }$ and $C _ { 2 }$ intersect (but do not touch) at exactly two points
(D) $C _ { 1 }$ and $C _ { 2 }$ neither intersect nor touch each other

jee-advanced 2008 Q4 Circle Identification and Classification View

Let $a$ and $b$ be non-zero real numbers. Then, the equation $$\left( a x ^ { 2 } + b y ^ { 2 } + c \right) \left( x ^ { 2 } - 5 x y + 6 y ^ { 2 } \right) = 0$$ represents
(A) four straight lines, when $c = 0$ and $a , b$ are of the same sign
(B) two straight lines and a circle, when $a = b$, and $c$ is of sign opposite to that of $a$
(C) two straight lines and a hyperbola, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$
(D) a circle and an ellipse, when $a$ and $b$ are of the same sign and $c$ is of sign opposite to that of $a$

jee-advanced 2008 Q7 Chord Length and Chord Properties View

A straight line through the vertex $P$ of a triangle $P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the triangle $P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then
(A) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 2 } { \sqrt { Q S \times S R } }$
(B) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 2 } { \sqrt { Q S \times S R } }$
(C) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 4 } { Q R }$
(D) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 4 } { Q R }$

jee-advanced 2008 Q12 Chord Length and Chord Properties View

Consider
$$\begin{aligned} & L _ { 1 } : 2 x + 3 y + p - 3 = 0 \\ & L _ { 2 } : 2 x + 3 y + p + 3 = 0 \end{aligned}$$
where $p$ is a real number, and $C : x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 30 = 0$. STATEMENT-1 : If line $L _ { 1 }$ is a chord of circle $C$, then line $L _ { 2 }$ is not always a diameter of circle $C$.
and
STATEMENT-2 : If line $L _ { 1 }$ is a diameter of circle $C$, then line $L _ { 2 }$ is not a chord of circle $C$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

jee-advanced 2008 Q15 Circle Equation Derivation View

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
The equation of circle $C$ is
(A) $\quad ( x - 2 \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(B) $( x - 2 \sqrt { 3 } ) ^ { 2 } + \left( y + \frac { 1 } { 2 } \right) ^ { 2 } = 1$
(C) $\quad ( x - \sqrt { 3 } ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
(D) $( x - \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$

jee-advanced 2009 Q22 Circle Equation Derivation View

Tangents drawn from the point $P ( 1,8 )$ to the circle
$$x ^ { 2 } + y ^ { 2 } - 6 x - 4 y - 11 = 0$$
touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $P A B$ is
(A) $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 19 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 4 x - 10 y + 19 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 29 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 19 = 0$

jee-advanced 2009 Q36 Circles Tangent to Each Other or to Axes View

The centres of two circles $C_{1}$ and $C_{2}$ each of unit radius are at a distance of 6 units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_{1}$ and $C_{2}$ and $C$ be a circle touching circles $C_{1}$ and $C_{2}$ externally. If a common tangent to $C_{1}$ and $C$ passing through $P$ is also a common tangent to $C_{2}$ and $C$, then the radius of the circle $C$ is

jee-advanced 2010 Q36 Circle-Related Locus Problems View

Tangents are drawn from the point $P ( 3,4 )$ to the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ touching the ellipse at points A and B.
The equation of the locus of the point whose distances from the point $P$ and the line AB are equal, is
A) $9 x ^ { 2 } + y ^ { 2 } - 6 x y - 54 x - 62 y + 241 = 0$
B) $x ^ { 2 } + 9 y ^ { 2 } + 6 x y - 54 x + 62 y - 241 = 0$
C) $9 x ^ { 2 } + 9 y ^ { 2 } - 6 x y - 54 x - 62 y - 241 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 2 x y + 27 x + 31 y - 120 = 0$

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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