Equation of Normal at Given Point

Q. The normal chord of a parabola $y^2=4ax$ at a point whose ordinate is equal to abscissa, subtends a right angle at

1. focus
2. vertex
3. ends of the latusrectum
4. any point on directrix

Q. Normal equations to parabola $y^2=4ax$ passing through point $(5a,2a)$ are

1. $y=-3x+17a$
2. $x=-3y+11a$
3. $y=x-3a$
4. $y=-2x+12a$

Q. The tangents and normals at the ends of the latus rectum of a parabola forms a

1. parallelogram
2. rectangle
3. square
4. rhombus

Q. If the tangents drawn from a point to the parabola $y^2=4ax$ are also normals to the parabola $x^2=4by$, then

1. $b^2\ge 8a^2$
2. $b^2\ge 2a^2$
3. $a^2\ge 2b^2$
4. $a^2\ge 8b^2$

Q.

It is a continuous function $f$ defined on the real line $R$, assume positive and negative values in $R$ then the equation $f\left(x\right)=0$ has root in $R$. For example, if it is known that a continuous function $f$ on $R$ is positive at some point and its minimum value is negative then the equation $f\left(x\right)=0$ has a root in $R$. Consider $f\left(x\right)=k{e}^x–x$ for all real $x$ where $k$ is a real constant. For$k>0$, the set of all values of $k$ for which $k{e}^x-x=0$ has two distinct roots is

1. $\left(0,\frac{1}{e}\right)$

2. $\left(\frac{1}{e},1\right)$

3. $\left(\frac{1}{e},\infty \right)$

4. $(0,1)$

Q. If the tangents and normals at the extremities of a focal chord of any standard parabola intersect at $S\equiv (x_1,y_1)$ and $R\equiv (x_2,y_2)$ respectively, then

1. If the equation of parabola is $y^2=\pm 4ax$, then $y_1=y_2$
2. If the equation of parabola is $x^2=\pm 4ay$, then $x_1=x_2$
3. the circle with focal chord as diameter will always pass through the points $R$ and $S$
4. the circle with $RS$ as diameter will always pass through extemities of focal chord

Q. If the tangents drawn from a point to the parabola $y^2=4ax$ are also normals to the parabola $x^2=4by$, then

1. $b^2\ge 8a^2$
2. $b^2\ge 2a^2$
3. $a^2\ge 9b^2$
4. $a^2\ge 8b^2$

Q. A normal drawn to parabola $y^2=4ax$ meet the curve again at $Q$ such that angle subtended by $PQ$ at vertex is ${90}^{\circ }$, then coordinates of $P$ can be

1. $(8a,4\sqrt{2}a)$
2. $(8a,4a)$
3. $(2a,-2\sqrt{2}a)$
4. $(2a,2\sqrt{2}a)$

Q. If the minimum distance between the parabolas $y^2-4x-8y+40=0$ and $x^2-8x-4y+40=0$ is $d$, then the value of $d^2$ is

Q. If the parabolas $y^2=4b(x-c)$ and $y^2=8ax$ have a common normal except the axis of symmetry of the parabolas, then the range of $\frac{c}{2a-b}$ is

1. $[0,\infty )$
2. $(2,\infty )$
3. $[-2,\infty )$
4. $[2,\infty )$

Q. Locus of a point through which three normals of parabola $y^2=4ax$ are passing, two of which are making angles $\alpha$ and $\beta$ with positive $x-$ axis and $\tan \alpha \cdot \tan \beta =2$ is

1. $y(y^2-2ax)=0$
2. $y(y^2+2ax)=0$
3. $y(y^2-4ax)=0$
4. $y(y^2-ax)=0$

Q.

Find the equation of normal to the parabola $y^2=4ax$ at point (h, k) on the parabola

1. $y-k=\frac{-k}{a(x-h)}$

2. $y-k=\frac{-k}{2a(x-h)}$

3. $y-k=\frac{k}{2a(x-h)}$

4. $y-k=\frac{-k}{2(x-h)}$

Q. Let $N$ be the normal on $y^2=4x$ at $S(1,2)$. A circle is inscribed on $SP$ as diameter, where $P$ is the focus of $y^2=4x$. If the length of the intercept made by the circle on $N$ is $k$, then the value of $k^4$ is

Q. Two tangents are drawn to the parabola $y^2=8x$ which meets the tangent at vertex at $P$ and $Q$ respectively. If $PQ=4\text{units}$, then the locus of the point of intersection of the two tangents is

1. $y^2=8(x+2)$
2. $y^2=8(x-2)$
3. $x^2=8(y+2)$
4. $x^2=8(y-2)$

Q. If the tangents and normals at the extremities of a focal chord of any standard parabola intersect at $S\equiv (x_1,y_1)$ and $R\equiv (x_2,y_2)$ respectively, then

1. If the equation of parabola is $y^2=\pm 4ax$, then $y_1=y_2$
2. If the equation of parabola is $x^2=\pm 4ay$, then $x_1=x_2$
3. the circle with focal chord as diameter will always pass through the points $R$ and $S$
4. the circle with $RS$ as diameter will always pass through extemities of focal chord

Q. If normals drawn to $y^2=12x$ makes an angle of $45°$ with $x-$axis, then foot of the normals is/are

1. $(12,-12)$
2. $(12,12)$
3. $(3,-6)$
4. $(3,6)$

Q. Length of normal chord of parabola $y^2=4x$ which makes an inclination of $\frac{\pi }{4}$ with the positive direction of $x-$axis is

1. $8$ units
2. $8\sqrt{2}$ units
3. $4$ units
4. $4\sqrt{2}$ units

Q. The normal to the curve $y(x-2)(x-3)=x+6$ at the point where the curve intersects the $y$-axis passes through the point:

1. $(-\frac{1}{2},-\frac{1}{2})$
2. $(\frac{1}{2},\frac{1}{2})$
3. $(\frac{1}{2},-\frac{1}{3})$
4. $(\frac{1}{2},\frac{1}{3})$

Q. The point on the parabola $y=x^2+7x+2$ which is closest to the line $y=3x-3$ is

1. $(2,20)$
2. $(1,10)$
3. $(-3,-28)$
4. $(-2,-8)$

Q. If $P$ and $Q$ are the points of contact of tangents drawn from the point $T$ to $y^2=4ax$ and $PQ$ be a normal of the parabola at $P$, then the locus of the point which bisects $TP$ is

1. $x+a=0$
2. $x+2a=0$
3. $x=0$
4. $x=1$

Q. If the normal to the parabola $y^2=4ax$ at the point $(a{t}^2,2at)$ cuts the parabola again at $(a{T}^2,2aT)$, then the range of $T$ is

1. $T\in (-2,2)$
2. $T\in (-\infty ,-8)\cup (8,\infty )$
3. $T\in (-2\sqrt{2},2\sqrt{2})$
4. $T\in [-\infty ,-2\sqrt{2}]\cup [2\sqrt{2},\infty ]$