Tangent and Normal to a Parabola

The slope of the normal to the parabola $y=\frac{x^2}{4}-2$ passing through the point $\left(10, -1\right)$ is

From a point $\left(C, 0\right)$ three normal are drawn to the parabola $y^2=x$ . Then

If two tangents are drawn from the point $\left(-1,-6\right)$ two tangents are drawn to the parabola $y^2=4x$. Then, the angle between two tangents is

Consider a parabola $y^2=4ax$, If the normals at points $P\left(t_1\right)$ and $Q\left(t_2\right)$ on the parabola intersect at $R$ on the curve then

If two tangents drawn from a point $P$ to the parabola $y^2=4x$ are at right angles, then the locus of $P$ is

The line $x+y=6$ is a normal to the parabola $y^2=8x$ at the point

If $y=mx+4$ is a tangent to both the parabolas, $\mathrm{}{y}^2=4x$ and $x^2=2by,$ then $b$ is equal to

If$2x+3y=\alpha , x-y=\beta$ and $kx+15y=r$ are $3$ concurrent normals of parabola $y^2=\lambda x$, then the value of $k$ is

The point on the parabola $y^2=8x$ at which the normal is inclined at ${60}^{°}$ to the $x$-axis has the coordinates

The equation of tangent with slope $\frac{2}{7}$ to Parabola $2x^2=7y$ is ......

If$2x+3y=\alpha , x-y=\beta$ and $kx+15y=r$ are $3$ concurrent normals of parabola $y^2=\lambda x$, then the value of $k$ is

The equation of the common tangent to the curves $y^2=4x$ and $x^2+32y=0$ is $x+by+c=0$. The value of $\left|{\sin }^{–1}\left(\sin 1\right)+{\sin }^{–1}\left(\sin b\right)+{\sin }^{–1}\left(\sin c\right)\right|$ is equal to

Tangents are drawn from a point $P$ to parabola $y^2=16x$ enclosing an angle of $45°$. Then, locus of point $P$ will be

Given a parabola $y^2=4ax,$ if line $y=3x-8$ touches the parabola at point $p\left(\alpha ,\beta \right),$ then $\left(\frac{18\alpha +\beta }{24-\beta }\right)=$

The point on the parabola $y^2=8x$ at which the normal is inclined at ${60}^{°}$ to the $x$-axis has the coordinates

Find the locus of the point $P$ if tangents $PQ$ and $PR$ are drawn to the parabolas $y^2=20\left(x+5\right)$ and $y^2=60\left(x+15\right)$ respectively, such that $\angle RPQ=\frac{\pi }{2}$

The equation of image of parabola $y^2=4x$ about the tangent of parabola $y^2=4x$ at $P(1, 2)$ is

The area of the triangle formed by tangents of the parabola $y^2=8x$ at the three points on the parabola having parameter $t_1, t_2$ and $t_3$ is

Number of common normals to the circle ${\left(x-5\right)}^2+y^2=9$ and the parabola $y^2=8x$, is

If the line $y=mx+c$ is tangent to the circle $x^2+y^2=5r^2$ and the parabola $y^2-4x-2y+4\lambda +1=0$ and point of contact of the tangent with the parabola is $\left(8,5\right)$, then find the value of $\left(25r^2+\lambda +2m-c\right)$.