Two vertices A and B of a triangle ABC are (-3,0) and (3,0) . The vertex C is varying in such a

step 1 In this problem of conic we have given that two vertices a and b so a and b are the two vertices

Two vertices A and B of a triangle ABC are (-3,0) and (3,0)...

Two vertices $\mathrm{A}$ and $\mathrm{B}$ of a triangle $\mathrm{ABC}$ are $(-3,0)$ and $(3,0) .$ The vertex $\mathrm{C}$ is varying in such a way that $4 \tan \frac{\mathrm{A}}{2} \tan \frac{\mathrm{B}}{2}=1$. The locus of $\mathrm{C}$ is (a) $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$ (b) $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ (c) $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ (d) $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$

Two vertices $\mathrm{A}$ and $\mathrm{B}$ of a triangle $\mathrm{ABC}$ are $(-3,0)$ and $(3,0) .$ The vertex $\mathrm{C}$ is varying in such a way that $4 \tan \frac{\mathrm{A}}{2} \tan \frac{\mathrm{B}}{2}=1$. The locus of $\mathrm{C}$ is
(a) $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$
(b) $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$
(c) $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$
(d) $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$

Key Concepts

Conic Sections

Conic sections, such as ellipses, are curves obtained by intersecting a plane with a right circular cone. In this problem, the derived algebraic equation from the trigonometric condition represents an ellipse, demonstrating how geometric constraints can yield specific conic sections as their loci.

Triangle Geometry in the Cartesian Plane

Triangle geometry examines the relationships among the vertices, sides, and angles of a triangle. When two vertices are fixed in the coordinate plane, as in this problem, the variation of the third vertex according to a given trigonometric condition provides a bridge between geometric configurations and their algebraic representations.

Tangent Half?Angle Identity

The tangent half-angle identity relates the tangent of half of an angle to other elements of a triangle, often simplifying complex angle relationships. By expressing the angles in terms of their half values, the identity allows one to convert angle conditions into algebraic equations, which are easier to handle in the coordinate setting.

Locus

In coordinate geometry, the locus of a point is the set of all points satisfying a specific condition or equation. In this context, the condition imposed on the vertex of the triangle defines a curve on the Cartesian plane, where every point on the curve meets the given trigonometric relation.

Transcript

Please give Ace some feedback