Ellipse
Tangent to Ellipse
Grade 11
Question:
<p>Consider the particle travelling clockwise on the elliptical path \(\frac{x^2}{100} + \frac{y^2}{25} = 1\). The particle leaves the orbit at the point (-8, 3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the y-axis?</p>
<p>(a) \(\left(0, \frac{25}{3}\right)\)</p>
<p>(b) \(\left(0, -\frac{25}{3}\right)\)</p>
<p>(c) \((0, 9)\)</p>
<p>(d) \(\left(0, \frac{7}{3}\right)\)</p>
Step-by-Step Solution
Key Concept: Find the equation of the tangent to the ellipse at a given point using the tangent formula, then find where it intersects the y-axis.
<p>The ellipse is \(\frac{x^2}{100} + \frac{y^2}{25} = 1\) with \(a^2 = 100\), \(b^2 = 25\). Find the tangent at point \((-8, 3)\): \(\frac{x(-8)}{100} + \frac{y(3)}{25} = 1\), which simplifies to \(\frac{-8x}{100} + \frac{3y}{25} = 1\) or \(-2x + 12y = 100\), i.e., \(-x + 6y = 50\) or \(x - 6y + 50 = 0\). At the y-axis, \(x = 0\): \(0 - 6y + 50 = 0\), so \(y = \frac{50}{6} = \frac{25}{3}\).</p>
Correct Answer: A