EXERCISES 12 (b)

Find the equation of the parabola of the following case: The axis is horizontal and the parabola passes through the points $(2,-1),(-2,-4)$ and $(-1,3)$.

Find the equation of tangents and normals to the parabola $y^2=4ax$ at the ends of its latus rectum.

Find the equations of tangents and normals to the parabola $y^2=4ax$ at the points where it is cut by the line $y=3x-a.$

Show that the tangent to the parabola $y^2=4ax$ at the point $\left(a^1, b^1\right)$ is perpendicular to the tangent at the point $\left(\frac{a^2}{a^1}, \frac{-4a^2}{b^1}\right).$

A tangent to the parabola $y^2=8x$ makes an angle $45°$ with the line $3x-y+5=0.$ Find the equation and the point of contact.

Prove that for all values of $k,$ the line $y=k\left(x+a\right)+\frac{a}{k}$ is a tangent to the parabola $y^2=4a\left(x+a\right)$.

Obtain the condition that the line $lx+my+n=0$ will touch the parabola $y^2=4ax.$

Prove that the line $4x-2y-1=0$ touches the parabola whose focus is at $\left(0, 0\right)$ and directrix is the line $y=2x-1.$