The question asks for the length of a chord cut by a line on a circle, properties of common chords between two circles, or optimization involving chord lengths.
If the midpoint of a chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$ is $( \sqrt { 2 } , 4 / 3 )$, and the length of the chord is $\frac { 2 \sqrt { \alpha } } { 3 }$, then $\alpha$ is : (1) 20 (2) 22 (3) 18 (4) 26
Let a circle $C$ pass through the points $( 4,2 )$ and $( 0,2 )$, and its centre lie on $3 x + 2 y + 2 = 0$. Then the length of the chord, of the circle $C$, whose mid-point is $( 1,2 )$, is : (1) $\sqrt { 3 }$ (2) $2 \sqrt { 2 }$ (3) $2 \sqrt { 3 }$ (4) $4 \sqrt { 2 }$
Q65. 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) $2 - \sqrt { 3 }$
Q67. Let the circle $C _ { 1 } : x ^ { 2 } + y ^ { 2 } - 2 ( x + y ) + 1 = 0$ and $C _ { 2 }$ be a circle having centre at $( - 1,0 )$ and radius 2 . If the line of the common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ intersects the $y$-axis at the point P , then the square of the distance of P from the centre of $\mathrm { C } _ { 1 }$ is : (1) 2 (2) 1 (3) 4 (4) 6
Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5:2$, then the minimum value of $\alpha + \beta$ is equal to: (A) 7 (B) 5 (C) 22 (D) 16
the parabola $\mathbf { y } ^ { 2 } = 8 \mathbf { x }$ such that $\left( \frac { 7 } { 3 } , \frac { 4 } { 3 } \right)$ is the centrodd of the $( B C ) ^ { 2 }$ is equal to (A) 120 (B) 150 (C) 90
On the coordinate plane, a circle with radius 12 intersects the line $x + y = 0$ at two points, and the distance between these two points is 8. If this circle intersects the line $x + y = 24$ at points $P$ and $Q$, then the length of segment $\overline { P Q }$ is $\_\_\_\_$ (14)$\sqrt { (15) }$. (Express as a simplified radical)
As shown in the figure, $L$ is a line passing through the origin $O$ on the coordinate plane, $\Gamma$ is a circle centered at $O$, and $L$ and $\Gamma$ have one intersection point $A ( 3,4 )$. It is known that $B , C$ are two distinct points on $\Gamma$ satisfying $\overrightarrow { B C } = \overrightarrow { O A }$. Select the correct options. (1) The other intersection point of $L$ and $\Gamma$ is $( - 4 , - 3 )$ (2) The slope of line $B C$ is $\frac { 3 } { 4 }$ (3) $\angle A O C = 60 ^ { \circ }$ (4) The area of $\triangle A B C$ is $\frac { 25 \sqrt { 3 } } { 2 }$ (5) $B$ and $C$ are in the same quadrant
On the coordinate plane, let $\Gamma$ be a circle with center at the origin, and $P$ be one of the intersection points of $\Gamma$ and the $x$-axis. It is known that the line passing through $P$ with slope $\frac{1}{2}$ intersects $\Gamma$ at another point $Q$, and $\overline{PQ} = 1$. Then the radius of $\Gamma$ is . (Express as a simplified radical)
On the coordinate plane, let $L _ { 1 }$ and $L _ { 2 }$ be two lines passing through point $(3, 1)$ with slopes $m$ and $- m$ respectively, where $m$ is a real number. Let $\Gamma$ be a circle with center at the origin. Given that $\Gamma$ intersects $L _ { 1 }$ at two distinct points $A$ and $B$, and the distance from the center to $L _ { 1 }$ is 1, and $\Gamma$ is tangent to $L _ { 2 }$, then the length of chord $\overline { A B }$ is (express as a fraction in lowest terms).
Let $a$ be a real number, and let $C$ be the circumference of the circle with center $(0,\, a)$ and radius $1$ in the coordinate plane.
[(1)] Find the range of $a$ such that $C$ is entirely contained in the region represented by the inequality $y > x^2$.
[(2)] Suppose $a$ is in the range found in (1). Let $S$ be the part of $C$ satisfying $x \geq 0$ and $y < a$. For a point $\mathrm{P}$ on $S$, let $L_{\mathrm{P}}$ be the length of the chord cut off from the tangent line to $C$ at $\mathrm{P}$ by the parabola $y = x^2$. Find the range of $a$ such that there exist two distinct points $\mathrm{Q}$, $\mathrm{R}$ on $S$ satisfying $L_{\mathrm{Q}} = L_{\mathrm{R}}$.
$ABCDEF$ is a regular hexagon $|AL| = |LB|$ $| \mathrm { BC } | = 4 \mathrm {~cm}$ $| \mathrm { DK } | = 3 \mathrm {~cm}$ $|KE| = \mathrm { x }$ According to the given information above, what is x in cm? A) $4 \sqrt { 3 }$ B) $3 \sqrt { 5 }$ C) $3 \sqrt { 7 }$ D) 6 E) 7
$ABCDEF$ is a regular hexagon $\mathrm { K } , \mathrm { L } \in [ \mathrm { AD } ]$ $| \mathrm { AB } | = 6$ units $| \mathrm { KL } | = \mathrm { x }$ In the figure, points $K$ and $L$ are on semicircles with diameters $AB$ and $DE$ respectively. Accordingly, what is $x$ in units? A) 5 B) 6 C) 9 D) $3 \sqrt { 3 }$ E) $6 \sqrt { 3 }$
Let $m$ and $n$ be real numbers. In the rectangular coordinate plane, a circle passing through point $A(4, 1)$ is drawn with equation $$x^{2} + y^{2} - 2x + 6y = n$$ The line $y = mx$ drawn in the plane intersects this circle at points $B$ and $C$. Given that $m(\widehat{BAC}) = 90^{\circ}$, what is the sum $m + n$? A) 8 B) 9 C) 10 D) 11 E) 12