Final answer:
To find the equations of the tangent lines to the given ellipse that pass through a given point, substitute the coordinates of the point into the equation of the ellipse to find the points of tangency. Then, use the point-slope form of a line to find the equations of the tangent lines.
Step-by-step explanation:
The equation of an ellipse is given by x^2/72 + y^2/9 = 1. To find the equations of the tangent lines to the ellipse that pass through the point (24, 3), we first need to find the points of tangency on the ellipse. We can do this by substituting the equation of the line passing through (24, 3) into the equation of the ellipse. This will give us two points of tangency. Once we have the points of tangency, we can find the equations of the tangent lines using the point-slope form of a line.
Step 1: Write down the equation of the ellipse in its standard form The given ellipse equation is: x^2 + 8y^2 = 72 We want to express this in the standard form of an ellipse equation: (x^2/a^2) + (y^2/b^2) = 1 By dividing every term by 72, we obtain: (x^2/72) + (y^2/9) = 1
Step 2: Find the general equation of any line passing through the given point A line passing through the point (24, 3) can be written as: y - y_1 = m(x - x_1) where (x_1, y_1) = (24, 3) and m is the slope. This gives us: y - 3 = m(x - 24)
Step 3: Write the condition for tangency For the line to be tangent to the ellipse, it must satisfy the ellipse's equation and intersect the ellipse at exactly one point.
The condition for tangency, when an ellipse is given by the equation (x^2/a^2) + (y^2/b^2) = 1 and a line is given by y = mx + c, is: b^2m^2 + a^2 = a^2b^2(c^2/b^2 - 1/(a^2m^2 + b^2)) Since we know that the line goes through the point (24, 3), by substituting y = 3 and x = 24 into our line equation, we can find the intercept c: 3 = m(24) + c c = 3 - 24m
Step 4: Solve for the slope(s) of the tangent line(s) Now, substituting the values of a^2 = 72, b^2 = 9, and c = 3 - 24m into the tangency condition, we obtain: 9m^2 + 72 = 72 * 9 * ((3 - 24m)^2/9 - 1/(72m^2 + 9)) After expanding and simplifying, this equation will give us the possible values of m.
Step 5: Write the equation(s) of the tangent line(s) Once we have solved for m (there may be two solutions, since a pair of tangents could exist), we can substitute each m back into the point-slope form of the line equation to get the equations of the tangent lines.