P(6,2 √(3)) is a point on (x^2)/(9)-(y^2)/(4)=1 . If Q is the projection of P on transverse axis

step 1 In this problem of conic section, we have given point P, whose coordinates are 6 and 2 root 3, i

P(6,2 √(3)) is a point on (x^2)/(9)-(y^2)/(4)=1 . If Q is...

$\mathrm{P}(6,2 \sqrt{3})$ is a point on $\frac{\mathrm{x}^{2}}{9}-\frac{\mathrm{y}^{2}}{4}=1 .$ If $\mathrm{Q}$ is the projection of $\mathrm{P}$ on transverse axis, the point of contact of the tangent drawn from $\mathrm{Q}$ to the auxiliary circle of given hyperbola is (a) $\left(\pm \frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)$ (b) $\left(\pm \frac{3}{2}, \pm \frac{3 \sqrt{3}}{2}\right)$ (c) $\left(\frac{3}{2}, \pm \frac{3 \sqrt{3}}{2}\right)$ (d) $\left(-\frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)$

$\mathrm{P}(6,2 \sqrt{3})$ is a point on $\frac{\mathrm{x}^{2}}{9}-\frac{\mathrm{y}^{2}}{4}=1 .$ If $\mathrm{Q}$ is the projection of $\mathrm{P}$ on transverse axis, the point of contact of the tangent drawn from $\mathrm{Q}$ to the auxiliary circle of given hyperbola is
(a) $\left(\pm \frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)$
(b) $\left(\pm \frac{3}{2}, \pm \frac{3 \sqrt{3}}{2}\right)$
(c) $\left(\frac{3}{2}, \pm \frac{3 \sqrt{3}}{2}\right)$
(d) $\left(-\frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)$

Key Concepts

Tangent to a Circle

A tangent to a circle is a line that touches the circle at exactly one point, and at that point, the tangent is perpendicular to the radius drawn to the point of contact. In problems involving tangents from an external point, the properties of the tangent line and its point of contact, including the conditions for tangency, are critical in forming geometric relationships and solving for unknown quantities.

Projection on an Axis

Projection of a point onto an axis involves finding the foot of the perpendicular from the point to that axis. In many geometric problems, such as those involving conic sections, projecting a point onto the transverse axis helps reduce complexity by focusing on the coordinate that directly influences the conic’s primary orientation.

Hyperbola

A hyperbola is a type of conic section that is defined as the set of all points in a plane such that the absolute difference of the distances from two fixed points (the foci) is constant. It is typically represented in a standard form equation, and its structure includes transverse and conjugate axes which determine its overall shape and orientation.

Auxiliary Circle

The auxiliary circle of a hyperbola is a circle that has the same center as the hyperbola and a radius equal to the length of the semi-transverse axis. This circle is used as a tool in analyzing the hyperbola, particularly when relating angular parameters or constructing tangents, by providing a simplified context where the properties of the hyperbola can be more directly interpreted.

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