Class 11

Math

Co-ordinate Geometry

Conic Sections

If the normal at one end of the latus rectum of the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ passes through one end of the minor axis, then prove that eccentricity is constant.

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Find the angle between the two tangents from the origin to the circle $(x−7)_{2}+(y+1)_{2}=25$

Tangent drawn from the point $P(4,0)$ to the circle $x_{2}+y_{2}=8$ touches it at the point $A$ in the first quadrant. Find the coordinates of another point $B$ on the circle such that $AB=4$ .

Let $P$ be a point on the circle $x_{2}+y_{2}=9,Q$ a point on the line $7x+y+3=0$ , and the perpendicular bisector of $PQ$ be the line $x−y+1=0$ . Then the coordinates of $P$ are $(0,−3)$ (b) $(0,3)$ $(2572 ,3521 )$ (d) $(−2572 ,2521 )$

If the distances from the origin of the centers of three circles $x_{2}+y_{2}+2λx−c_{2}=0,(i=1,2,3),$ are in GP, then prove that the lengths of the tangents drawn to them from any point on the circle $x_{2}+y_{2}=c_{2}$ are in GP.

If the circle $x_{2}+y_{2}+2gx+2fy+c=0$ bisects the circumference of the circle $x_{2}+y_{2}+2g_{prime}x+2f_{prime}y+c_{prime}=0$ then prove that $2g_{prime}(g−g_{prime})+2f_{prime}(f−f_{prime})=c−c_{′}$

Tangents $PAandPB$ are drawn to $x_{2}+y_{2}=a_{2}$ from the point $P(x_{1},y_{1})˙$ Then find the equation of the circumcircle of triangle $PAB˙$

Find the equations to the common tangents of the circles $x_{2}+y_{2}−2x−6y+9=0$ and $x_{2}+y_{2}+6x−2y+1=0$

Find the equation of the circle whose radius is 3 and which touches internally the circle $x_{2}+y_{2}−4x−6y=−12=0$ at the point $(−1,−1)˙$