The foci of a hyperbola are and its eccentricity is . A tangent, perpendicular to the line , is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the -

Question

The foci of a hyperbola are  and its eccentricity is . A tangent, perpendicular to the line , is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the - and -axes are  and  respectively, then  is equal to .

Canvas Classes · Accepted Answer

🧠 Tangent Slope From Perpendicularity, Then Use the Slope-Tangent Formula

The problem gives a line , which has slope . The tangent we want is perpendicular to this, so its slope is .

For a hyperbola, the tangent of slope  has the form  — this is the standard "slope form" tangent. It bypasses parametric coordinates entirely. So once you know  and , the tangent line drops out in one line.

The "first quadrant" condition picks one of the two parallel tangents (the  sign). After that, find the in

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