LFM Pure

jee-main 2023 Q71 Tangent Lines and Tangent Lengths View

Let the tangents at the points $A ( 4 , - 11 )$ and $B ( 8 , - 5 )$ on the circle $x ^ { 2 } + y ^ { 2 } - 3 x + 10 y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to
(1) $\frac { 3 \sqrt { 3 } } { 4 }$
(2) $2 \sqrt { 13 }$
(3) $\sqrt { 13 }$
(4) $\frac { 2 \sqrt { 13 } } { 3 }$

jee-main 2023 Q71 Inscribed/Circumscribed Circle Computations View

A triangle is formed by the tangents at the point $( 2,2 )$ on the curves $y ^ { 2 } = 2 x$ and $x ^ { 2 } + y ^ { 2 } = 4 x$, and the line $\mathrm { x } + \mathrm { y } + 2 = 0$. If $r$ is the radius of its circumcircle, then $r ^ { 2 }$ is equal to $\_\_\_\_$

jee-main 2023 Q75 Circle Identification and Classification View

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $\alpha$ and the number of persons who speaks only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^{2}x^{2} + \alpha^{2}y^{2}\right) = \alpha^{2}\beta^{2}$ is
(1) $\frac{\sqrt{119}}{12}$
(2) $\frac{\sqrt{117}}{12}$
(3) $\frac{3\sqrt{15}}{12}$
(4) $\frac{\sqrt{129}}{12}$

jee-main 2023 Q78 Circle-Line Intersection and Point Conditions View

The number of integral values of $k$ for which the line $3x + 4y = k$ intersects the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ at two distinct points is $\_\_\_\_$.

jee-main 2023 Q86 Tangent Lines and Tangent Lengths View

Let a common tangent to the curves $y^2 = 4x$ and $(x-4)^2 + y^2 = 16$ touch the curves at the points $P$ and $Q$. Then $PQ^2$ is equal to $\_\_\_\_$.

jee-main 2024 Q65 Intersection of Circles or Circle with Conic View

If the circles $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$ and $x ^ { 2 } + y ^ { 2 } - 4 x - 4 y + 4 = 0$ intersect at exactly two distinct points, then
(1) $5 < \mathrm { r } < 9$
(2) $0 < \mathrm { r } < 7$
(3) $3 < r < 7$
(4) $\frac { 1 } { 2 } < \mathrm { r } < 7$

jee-main 2024 Q65 Circle Equation Derivation View

If one of the diameters of the circle $x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 13 = 0$ is a chord of another circle $C$, whose center is the point of intersection of the lines $2 x + 3 y = 12$ and $3 x - 2 y = 5$, then the radius of the circle $C$ is
(1) $\sqrt { 20 }$
(2) 4
(3) 6
(4) $3 \sqrt { 2 }$

jee-main 2024 Q65 Inscribed/Circumscribed Circle Computations View

Let $\left( 5 , \frac { a } { 4 } \right)$, be the circumcenter of a triangle with vertices $A ( a , - 2 ) , B ( a , 6 )$ and $C \left( \frac { a } { 4 } , - 2 \right)$. Let $\alpha$ denote the circumradius, $\beta$ denote the area and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is
(1) 60
(2) 53
(3) 62
(4) 30

jee-main 2024 Q65 Chord Length and Chord Properties View

Let $C$ be a circle with radius $\sqrt { 10 }$ units and centre at the origin. Let the line $x + y = 2$ intersects the circle C at the points P and Q . Let MN be a chord of C of length 2 unit and slope - 1 . Then, a distance (in units) between the chord PQ and the chord MN is
(1) $3 - \sqrt { 2 }$
(2) $\sqrt { 2 } + 1$
(3) $\sqrt { 2 } - 1$
(4) $2 - \sqrt { 3 }$

jee-main 2024 Q66 Circle-Related Locus Problems View

Let the locus of the mid points of the chords of circle $x^2 + (y-1)^2 = 1$ drawn from the origin intersect the line $x + y = 1$ at $P$ and $Q$. Then, the length of $PQ$ is:
(1) $\frac{1}{\sqrt{2}}$
(2) $\sqrt{2}$
(3) $\frac{1}{2}$
(4) 1

jee-main 2024 Q66 Inscribed/Circumscribed Circle Computations View

A circle is inscribed in an equilateral triangle of side of length 12 . If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m + n ^ { 2 }$ is equal to
(1) 408
(2) 414
(3) 396
(4) 312