Conic Sections

Let an ellipse E: x²/a² + y²/b² = 1, a² > b², passes through (√3/2, 1) and has eccentricity 1/√3. If a circle, centered at focus F(α, 0), α > 0, of E and radius 2/√3 intersects E at two points P and Q, then PQ² is equal to:

Let the normal at a point $P$ $$ on the curve ${}^{}{}^{}$ $y^2-3x^2+y+10=0$ intersect the $$ $y-axis$ at $\left(0,\frac{3}{2}\right)$ $\left(\frac{}{}\right)$ . If $m$ $$ is the slope of the tangent at $P$ $$ to the curve, then $$ $\left|m\right|$ is equal to(2D)

Let the line $y=mx$ $$ and the ellipse $2x^2+y^2=1$ ${}^{}{}^{}$ intersect at a point $$ $P$ in the first quadrant. If the normal to this ellipse at $P$ $$ meets the co - ordinate axes at $\left(\frac{}{\text{exception 25:}}\right)$ $\left(-\frac{1}{3\sqrt[]{2}},0\right)$ and $$ $(0,\beta )$ then $\beta$ $$ is equal to:

The locus of a point which divides the line segment joining the point $(0,-1)$ $$ and a point on the parabola, $x^2=4y$ ${}^{}$ , internally in the ratio $1:2$ $$ , is

If the line y=mx+c is a common tangent to the hyperbola x²/100 - y²/64 = 1 and the circle x²+y²=36, then which one of the following is true?

For $a>0$ $$ , let the curves $C_1:y^2=ax$ ${}_{}{}^{}$ and $C_2:x^2=$ ${}_{}{}^{}$ ay intersect at origin $O$ $$ and a point $$ $P$ . Let the line $x=b(0<b<c)$ $$ intersect the chord $OP$ $$ and the $x$ $$ - axis at points $$ $Q$ and $R$ $$ , respectively. If the line $x=b$ $$ bisects the area bounded by the curves, $C_1$ ${}_{}$ and ${}_{}$ , and the area of $\frac{}{}$ $?\mathrm{O}\mathrm{Q}\mathrm{R}=\frac{1}{2}$ , then 'a' satisfies the equation:

If $3x+4y=12\sqrt[]{2}$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in R$ , then the distance between the foci of the ellipse is:

Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is perpendicular to both the line segments CP and CQ, then the set {P, Q} is equal to:

A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of any ellipse with eccentricity 1/3 and the distance of the nearer focus from this directrix is 8/√53, then the equation of the other directrix can be:

Let y = y(x) be solution of the differential equation logₑ(dy/dx) = 3x + 4y, with y(0) = 0.
If y(-2/3 logₑ2) = α logₑ2, then the value of α is equal to: