Class 12

Math

Co-ordinate Geometry

Conic Sections

$P(a,5a)$ and $Q(4a,a)$ are two points. Two circles are drawn through these points touching the axis of y. Centre of these circles are at

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A tangent is drawn to the ellipse to cut the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ and to cut the ellipse $c_{2}x_{2} +d_{2}y_{2} =1$ at the points P and Q. If the tangents are at right angles, then the value of $(c_{2}a_{2} )+(d_{2}b_{2} )$ is

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 coordinates of the focus, axis of the parabola ,the equation of directrix and the length of the latus rectum for $x_{2}=−9y$

The auxiliary circle of a family of ellipses passes through the origin and makes intercepts of 8 units and 6 units on the x and y-axis, respectively. If the eccentricity of all such ellipses is $21 ,$ then find the locus of the focus.

If the normal at any point $P$ on the ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ meets the axes at $Gandg$ respectively, then find the raio $PG:Pg=$ (a) $a:b$ (b) $a_{2}:b_{2}$ (c) $b:a$ (d) $b_{2}:a_{2}$

If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then find the eccentricity of the ellipse.

A man running a race course notes that the sum of the distance from the two flag posts from him is always $10$ m and the distance between the flag posts is $8$ m. Find the equation of the posts traced by the man.

An ellipse is drawn with major and minor axis of length $10$ and $8$ respectively. Using one focus a centre, a circle is drawn that is tangent to ellipse, with no part of the circle being outside the ellipse. The radius of the circle is (A) $3 $ (B) $2$ (C) $22 $ (D) $5 $