Class 11

Math

Co-ordinate Geometry

Conic Sections

Find the point on the ellipse $16x_{2}+11y_{2}=256$ where the common tangent to it and the circle $x_{2}+y_{2}−2x=15$ touch.

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Perpendiculars are drawn, respectively, from the points $PandQ$ to the chords of contact of the points $QandP$ with respect to a circle. Prove that the ratio of the lengths of perpendiculars is equal to the ratio of the distances of the points $PandQ$ from the center of the circles.

The line $3x+6y=k$ intersects the curve $2x_{2}+3y_{2}=1$ at points $AandB$ . The circle on $AB$ as diameter passes through the origin. Then the value of $k_{2}$ is__________

Find the equation of the smallest circle passing through the intersection of the line $x+y=1$ and the circle $x_{2}+y_{2}=9$

Statement 1 : Let $S_{1}:x_{2}+y_{2}−10x−12y−39=0,$ $S_{2}x_{2}+y_{2}−2x−4y+1=0$ and $S_{3}:2x_{2}+2y_{2}−20x=24y+78=0.$ The radical center of these circles taken pairwise is $(−2,−3)˙$ Statement 2 : The point of intersection of three radical axes of three circles taken in pairs is known as the radical center.

A variable circle passes through the point $A(a,b)$ and touches the x-axis. Show that the locus of the other end of the diameter through $A$ is $(x−a)_{2}=4by˙$

Find the number of integral values of $λ$ for which $x_{2}+y_{2}+λx+(1−λ)y+5=0$ is the equation of a circle whose radius does not exceed 5.

If one end of the a diameter of the circle $2x_{2}+2y_{2}−4x−8y+2=0$ is (3, 2), then find the other end of the diameter.

If $a>2b>0,$ then find the positive value of $m$ for which $y=mx−b1+m_{2} $ is a common tangent to $x_{2}+y_{2}=b_{2}$ and $(x−a)_{2}+y_{2}=b_{2}˙$