Class 12

Math

Co-ordinate Geometry

Conic Sections

A variable parabola $y_{2}=4ax,a$ (where $a=−41 )$ being the parameter, meets the curve $y_{2}+x−2=0$ at two points. The locus of the point of intersecion of tangents at these points is

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Let E1 and E2, be two ellipses whose centers are at the origin.The major axes of E1 and E2, lie along the x-axis and the y-axis, respectively. Let S be the circle $x_{2}+(y−1)_{2}=2$. The straight line x+ y =3 touches the curves S, E1 and E2 at P,Q and R, respectively. Suppose that $PQ=PR=322 $.If e1 and e2 are the eccentricities of E1 and E2, respectively, then the correct expression(s) is(are):

Find the equation of the circle with radius $5$ whose centre lies on $x$-axis and passes through the point $(2,3)$

Find the equation of the hyperbola satisfying the give conditions: Vertices $(0,±5)$ foci $(0,±8)$

Find the normal to the ellipse $18x_{2} +8y_{2} =1$ at point (3, 2).

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $36x_{2} +16y_{2} =1.$

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 $

Suppose that the foci of the ellipse $9x_{2} +5y_{2} =1$ are $(f_{1},0)and(f_{2},0)$ where $f_{1}>0andf_{2}<0.$ Let $P_{1}andP_{2}$ be two parabolas with a common vertex at (0, 0) and with foci at $(f_{1}.0)and$ (2f_2 , 0), respectively. Let$T_{1}$ be a tangent to $P_{1}$ which passes through $(2f_{2},0)$ and $T_{2}$ be a tangents to $P_{2}$ which passes through $(f_{1},0)$ . If $m_{1}$ is the slope of $T_{1}$ and $m_{2}$ is the slope of $T_{2},$ then the value of $(m121 +m22)$ is

Find the locus of point $P$ such that the tangents drawn from it to the given ellipse $a_{2}x_{2} +b_{2}y_{2} =1$ meet the coordinate axes at concyclic points.