Understanding the Equation of Tangent to Ellipse

The problem asks for the equation of the tangent line to a standard ellipse at a specific point on the ellipse. The standard equation of an ellipse centered at the origin is given by \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\). We are given a point \((x', y')\) which lies on this ellipse, and we need to find the equation of the line that is tangent to the ellipse at this particular point.

There is a standard method to find the equation of a tangent to a general quadratic curve (like an ellipse, parabola, or hyperbola) at a given point \((x', y')\) on the curve. This method involves replacing terms in the equation as follows:

• Replace \(x^2\) with \(xx'\)
• Replace \(y^2\) with \(yy'\)
• Replace \(x\) with \(\frac{x+x'}{2}\)
• Replace \(y\) with \(\frac{y+y'}{2}\)
• Replace \(xy\) with \(\frac{xy'+yx'}{2}\)

Applying the Formula for Equation of Tangent to Ellipse

Let's apply this method to the given equation of the ellipse: \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\).

We need to replace \(x^2\) with \(xx'\) and \(y^2\) with \(yy'\). The equation becomes:

$$ \frac{{xx'}}{{{a^2}}} + \frac{{yy'}}{{{b^2}}} = 1 $$

This equation represents the line that is tangent to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) at the point \((x', y')\). This is a standard formula for the equation of tangent to an ellipse.

Verifying the Equation of Tangent to Ellipse

Now, let's compare our derived equation with the given options:

• Option 1: \(\frac{{xx'}}{{{a^2}}} + \frac{{yy'}}{{{b^2}}} = 1\)
• Option 2: \(\frac{{x - x'}}{a} + \frac{{y - y'}}{b} = 1\)
• Option 3: \(\frac{{x(x - x')}}{{{a^2}}} + \frac{{y(y - y')}}{{{b^2}}} = 1\)
• Option 4: \(\frac{{{{(x')}^2}}}{{{a^2}}} + \frac{{{{(y')}^2}}}{{{b^2}}} = 1\)

Our derived equation, \(\frac{{xx'}}{{{a^2}}} + \frac{{yy'}}{{{b^2}}} = 1\), matches Option 1 exactly.

Option 4 is the equation that states the point \((x', y')\) lies on the ellipse, as it satisfies the ellipse equation itself. This is not the equation of the tangent line.

The formula \(\frac{{xx'}}{{{a^2}}} + \frac{{yy'}}{{{b^2}}} = 1\) is a fundamental result in coordinate geometry for finding the equation of tangent to an ellipse at a given point \((x', y')\) on it. This derivation method, often called the T=0 method, is a quick way to get the equation of tangent for quadratic curves.

Alternatively, one could use calculus by finding the derivative \(\frac{dy}{dx}\) implicitly from the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), evaluating the slope at \((x', y')\), and then using the point-slope form of a line \(y - y' = m(x - x')\). This calculus method also leads to the same equation of tangent.

Conclusion

The correct equation of the tangent at the point \((x', y')\) to the ellipse \(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\) is \(\frac{{xx'}}{{{a^2}}} + \frac{{yy'}}{{{b^2}}} = 1\). This is a standard formula derived using the property of tangents to conic sections.