class 11

Math

Co-ordinate Geometry

Conic Sections

Circle(s) touching x-axis at a distance 3 from the origin and having an intercept of length $27 $ on y-axis is (are)

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If $ω$ is one of the angles between the normals to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ at the point whose eccentric angles are $θ$ and $2π +θ$ , then prove that $sin2θ2cotω =1−e_{2} e_{2} $

Tangents are drawn from the points on the line $x−y−5=0$ to $x_{2}+4y_{2}=4$ . Then all the chords of contact pass through a fixed point. Find the coordinates.

Find the angle between the asymptotes of the hyperbola $16x_{2} −9y_{2} =1$ .

Find the equation for the ellipse that satisfies the given conditions: $b=3,ae=4$ centre at the origin; foci on the $x$- axis.

$AOB$ is the positive quadrant of the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ in which $OA=a,OB=b$ . Then find the area between the arc $AB$ and the chord $AB$ of the ellipse.

If the normals to the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ at the points $(X_{1},y_{1}),(x_{2},y_{2})and(x_{3},y_{3})$ are concurrent, prove that $∣∣ x_{1}x_{2}x_{3} y_{1}y_{2}y_{3} x_{1}y_{1}x_{2}y_{2}x_{3}y_{3} ∣∣ =0$.

Find the equation for the ellipse that satisfies the given conditions: Centre at $(0,0)$, major axis on the $y$-axis and passes through the points $(3,2)$ and $(1,6).$

An ellipse has $OB$ as the semi-minor axis, $FandF_{′}$ as its foci, and $∠FBF_{′}$ a right angle. Then, find the eccentricity of the ellipse.