The abscissa of the point on the curve √(x y)=a+x, the tangent at which cuts off equal intersect

step 1 in this problem let us differentiate the given equation we get 1 by 2 root x because differentia

The abscissa of the point on the curve √(x y)=a+x, the...

The abscissa of the point on the curve $\sqrt{x y}=a+x$, the tangent at which cuts off equal intersects from the coordinate axes is $(a>0)$
(a) $\frac{a}{\sqrt{2}}$
(b) $-\frac{a}{\sqrt{2}}$
(c) $a \sqrt{2}$
(d) $-a \sqrt{2}$

Key Concepts

Equal Intercepts Condition

The condition that a line cuts off equal intercepts from the coordinate axes imposes a specific relationship between the slope of the line and its intercepts. This condition is used to derive equations that relate the geometric properties of a curve and the properties of its tangents, ultimately helping to pinpoint specific points of tangency.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are intermingled. It allows us to compute the derivative without explicitly solving for one variable in terms of the other, which is particularly useful for curves defined by equations that are not easily separable.

Tangent Line to a Curve

The tangent line at a given point on a curve is the straight line that best approximates the curve near that point. Its slope is given by the derivative of the curve at that point, and the equation of the tangent line can be derived using the point-slope form, which is fundamental in understanding local linear approximation.

Intercepts of a Line

Intercepts are the points at which a line crosses the coordinate axes. The x-intercept is found by setting y to zero, and the y-intercept by setting x to zero. These intercepts are often used to understand the geometric characteristics of the line, and in certain problems, conditions on these intercepts can lead to important relationships.

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