The line y = x + 8 touches the cllipse x`^2`/a`^2` + y`^2`/b`^2` = 1 whose eccentricity is `sqrt2`/3. Find a and b.

For what value of p,y = x + p touches the ellipse 2x ^2 + 3y ^2 = 1 ?

Show that the straight line 3x + 4y = 20 touches the circle x ^2 + y ^2 = 16.

Find frac(d^2y)(dx^2) : (i) x^2/a^2+y^2/b^2 = 1

Find the equation of the tangents to the ellipse 3x ^2 + 4y ^2 = 12 which are perpendicular to the line 2x +y - 4 = 0.

Find the value of k for which the line x + 2y + k = 0 touches the circle x ^2 + y ^2 = 5.

Find the eccentricity of the hyperbola (ii) x ^2 /16 - y ^2 /9 = 1

For what value of c, the straight line 3x + y - c = 0 touches the circle x ^2 + y ^2 + 2x - 6y + 6 = 0 ?

Find the slope of tangent to the curve y = 3x^2 - 4x at point whose x - coordinate is 2.

Find the equations of the tangents to the hyperbola x ^2 - 2y ^2 = 18 which are perpendicular to the line x - y = 0.