Problem 15 – Mathematics (Analytic Geometry) The equation of a given conics is x2 + y2 = 25. Find the length of the sub-tangent at (-3, 4)

The equation of the given conic is x² + y² = 25, which is the equation of a circle with radius 5.

The sub-tangent of a point on a circle is the line segment from the point of tangency to the x-intercept of the tangent line.

The derivative of the equation of the circle is 2x + 2y*y' = 0. Solving for y', we get y' = -x/y.

Substituting the point (-3, 4) into y', we get y' = 3/4.

The equation of the tangent line at the point (-3, 4) is y - 4 = 3/4 * (x + 3).

Setting y = 0 to find the x-intercept, we get x = -16/3.

The length of the sub-tangent is the distance from the point (-3, 4) to the x-intercept (-16/3, 0), which can be found using the distance formula.

The distance is sqrt((x2 - x1)² + (y2 - y1)²) = sqrt(((-16/3) - (-3))² + (0 - 4)²) = sqrt((1/3)² + 4²) = sqrt(1/9 + 16) = sqrt(145/9) = 5/3 units.

So, the length of the sub-tangent at the point (-3, 4) is 5/3 units.