The tangent at a point on 16 x^2+25 y^2=400 meets the tangents at the ends of the major axis at

step 1 So in this question the tangent at the point on the ellipse meet the tangent at the ends of majo

The tangent at a point on 16 x^2+25 y^2=400 meets the...

The tangent at a point on $16 \mathrm{x}^{2}+25 \mathrm{y}^{2}=400$ meets the tangents at the ends of the major axis at $\mathrm{T}_{1}$ and $\mathrm{T}_{2} .$ Then the equation of the circle with on $\mathrm{T}_{1} \mathrm{~T}_{2}$ as diameter passes through the point (a) $(1,0)$ (b) $(3,0)$ (c) $(0,3)$ (d) $(0,-3)$

Key Concepts

Major Axis

The major axis of an ellipse is the longest diameter that runs through its center and both foci, determining the ellipse's maximum width. It establishes the orientation of the ellipse and helps in identifying the vertices, which become reference points for constructing associated tangents and other geometric elements.

Circle Defined by a Diameter

A circle defined with a given diameter is the set of all points that are equidistant from the midpoint of the diameter. This construction is based on the Thales' theorem, which states that any angle formed by a point on the circle, with the diameter as one side, is a right angle. Recognizing this property allows one to analyze geometric relationships and intersections involving circles and other curves.

Ellipse

An ellipse is a conic section defined as the set of points for which the sum of the distances to two fixed points (the foci) is constant. Its standard form equation highlights the lengths of the semi-major and semi-minor axes, and it demonstrates symmetry about both the major and minor axes, which are essential in analyzing its geometric properties.

Tangent to an Ellipse

A tangent line to an ellipse touches the ellipse at exactly one point and is perpendicular to the radius vector at the point of contact when considered in an affine geometry sense. The equation of the tangent line can often be derived using implicit differentiation or by employing the standard tangent form, and it plays a crucial role in problems involving intersections with other lines or curves.

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