The line \(x + 3y = 0\) is a diameter of the circle \(x^2 + y^2 + 6x + 2y = 0\).
False
The shortest distance from the point \((2, -7)\) to the circle \(x^2 + y^2 - 14x - 10y - 151 = 0\) is equal to 5.
False
If the line \(lx + my = 1\) is a tangent to the circle \(x^2 + y^2 = a^2\), then the point \((l,m)\) lies on a circle.
True
The point \((1,2)\) lies inside the circle \(x^2 + y^2 - 2x + 6y + 1 = 0\).
False
The line \(lx + my + n = 0\) will touch the parabola \(y^2 = 4ax\) if \(ln = am^2\).
True
If P is a point on the ellipse \(\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1\) whose foci are S and S′, then \(PS + PS′ = 8\).
False
The line \(2x + 3y = 12\) touches the ellipse \(\dfrac{x^2}{9} + \dfrac{y^2}{4} = 2\) at the point \((3,2)\).
True
The locus of the point of intersection of lines \(\sqrt{3}x - y - 4\sqrt{3}k = 0\) and \(\sqrt{3}kx + ky - 4\sqrt{3} = 0\) for different values of k is a hyperbola whose eccentricity is 2.
True