Question

Show that the line 3x – 4y + 10 = 0 is a tangent to the hyperbola x² – 4y² = 20. Also, find the point of contact.

Answer 1

Given equation of the hyperbola is x² – 4y² = 20

\(\frac {x^2}{20} - \frac {y^2}{5} = 1\)

Comparing this equation with \(\frac {x^2}{a^2} - \frac {y^2}{b^2} = 1\),

we get a² = 20 and b² = 5

Given equation of line is 3x – 4y + 10 = 0.

y = 3x/4 + 5/2

Comparing this equation with y = mx + c, we get

m = 3/4 and c = 5/2

For the line y = mx + c to be a tangent to the hyperbola \(\frac {x^2}{a^2} - \frac {y^2}{b^2} = 1\), we must have