Any tangent at a point P(x, y) to the ellipse (x^2)/(8)+(y^2)/(18)=1 meets the co-ordinate axes

step 1 Hello everyone. From the question first we are going to draw the diagram. We are given that any

Any tangent at a point P(x, y) to the ellipse...

Any tangent at a point $P(x, y)$ to the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{18}=1$ meets the co-ordinate axes in the points $A$ and $B$ such that the area of the triangle $O A B$ is least, then the point $P$ is
(a) $(\sqrt{8}, 0)$
(b) $(0, \sqrt{18})$
(c) $(2,3)$
(d) None

Key Concepts

Intercept Form of a Line

The intercept form of a line expresses the line equation based on its x-intercept and y-intercept. It is written as x/a + y/b = 1, where 'a' and 'b' are the x-intercept and y-intercept respectively. This form is particularly useful in geometric problems that involve finding areas of figures formed by the coordinate axes because it directly relates the intercepts to the triangle's dimensions.

Area Optimization in Geometry

Area optimization involves finding the extrema (minimum or maximum) of the area of a geometric figure under given constraints. This typically requires expressing the area in terms of one or more variables and then applying calculus techniques, such as differentiation, to determine the critical points. Such methods are common in problems where geometric figures depend on variable parameters.

Ellipse

An ellipse is a type of conic section defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. Ellipses have a standard equation and exhibit key properties such as axes of symmetry, vertices, and foci, which are useful in analyzing geometric relationships and behaviors in various problems.

Tangent to a Conic Section

A tangent to a conic section is a line that touches the curve at exactly one point without crossing it. For an ellipse, the tangent line at a given point can be derived using geometric properties or implicit differentiation. Studying tangents is important because they provide insights into the curve's local behavior and are instrumental in solving optimization and interception problems.

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