class 11

Math

Co-ordinate Geometry

Conic Sections

The circle passing through (1, -2) and touching the axis of x at (3, 0) also passes through the point

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Find the equation of the circle passing through the points $(4,1)$ and $(6,5)$ and whose centre is on the line $4x+y=16$

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.

Tangents are drawn from the points on a tangent of the hyperbola $x_{2}−y_{2}=a_{2}$ to the parabola $y_{2}=4ax˙$ If all the chords of contact pass through a fixed point $Q,$ prove that the locus of the point $Q$ for different tangents on the hyperbola is an ellipse.

Find the point $(α,β)$ on the ellipse $4x_{2}+3y_{2}=12,$ in the first quadrant, so that the area enclosed by the lines $y=x,y=β,x=α$ , and the x-axis is maximum.

Find the equation of a chord of the ellipse $25x_{2} +16y_{2} =1$ joining two points $P(4π )$ and $Q(45π )$

The tangent at a point $P$ on an ellipse intersects the major axis at $T,andN$ is the foot of the perpendicular from $P$ to the same axis. Show that the circle drawn on $NT$ as diameter intersects the auxiliary circle orthogonally.

Find the equation of the locus of the middle points of the chords of the hyperbola $2x_{2}−3y_{2}=1,$ each of which makes an angle of $45_{0}$ with the x-axis.

If the focal distance of an end of the minor axis of an ellipse (referred to its axes as the axes of $xandy$ , respectively) is $k$ and the distance between its foci is $2h,$ them find its equation.