Equation of Normal at Given Point

Q. The equations of the circles which touch the y-axis at the origin and the line 5x + 12y – 72 = 0 is

1. + + 2y = 0, + - 18y = 0
2. + - 6y = 0, + + 24y = 0
3. + + 18x = 0, + - 8x = 0
4. + + 4x = 0, + - 16x = 0

Q. The normal at $P(2,4)$ to $y^2=8x$ meets the parabola at $Q.$ Then the equation of the circle having normal chord $PQ$ as diameter is

1. $x^2+y^2-20x+8y-12=0$
2. $x^2+y^2-10x+4y-8=0$
3. $x^2+y^2-12x+6y-15=0$
4. $x^2+y^2-10x+8y-12=0$

Q.

Find the values of K for which the line $(K-3)x-(4-K^2)y+K^2-7K+6=0$ is

(a) Parallel to the x-axis

(b) Parallel to the y-axis,

(c) Passing through the origin.

Q. The length of normal chord of parabola $y^2=4x$, which subtends an angle of ${90}^{\circ }$ at the vertex is

1. $6\sqrt{3}\text{units}$
2. $7\sqrt{2}\text{units}$
3. $8\sqrt{3}\text{units}$
4. $9\sqrt{2}\text{units}$

Q.

If normal at p to parabola $y^2=4ax$ meets parabola again at Q such that PQ subtends a right angle at the vertex of $y^2=4ax$ then p will be

1. $(2a,2(\sqrt{2}a)$

2. $(-2a,2(\sqrt{2}a)$

3. $\left(2\right(\sqrt{2}a,2a)$

4. $\left(2\sqrt{2}a\right),-2a)$

Q. The normal to the parabola $y^2=8x$ at the point (2, 4) meets the parabola again at the point _____.

1. (18, -12)
2. (-18, 12)
3. (18, 12)
4. (-18, -12)

Q.

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

1. 

2. 

3. 

4. 

Q. The normal at $3$ points $P,Q,R$ on $y^2=4ax$ meet at a point $N.$ If $S$ is focus of parabola then the value of$\frac{SP+SQ+SR+SA}{MN}$ is
(where $A$ is vertex of parabola $M$ is the foot of perpendicular from $N$ on to tangent at vertex)

Q. Let $x=my+c$ is normal to $x^2=4y$, if $k^2+mk+m=0$ has only one real value of $k$, then value(s) of $c$ is/are

1. $0$
2. $72$
3. $-72$
4. $64$

Q. Prove that the normal chord at the point whose ordinate is equal to its abscissa subtends a right angle at the focus.

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. $\begin{array}{cc}Column1& Column2\\ \text{a. Tangents are drawn from the point}(2,3)& \text{p.}(9,-6)\\ \text{to the parabola}{y}^2=4x\text{then points of contact are}& \\ \text{b. From a point}P\text{on the circle}{x}^2+y^2=5,& \\ \text{the equation of chord of contact to the parabola}& \\ y^2=4x\text{is}y=2(x-2),\text{then the coordinate of}& \\ \text{point}P\text{will be}& \text{q.}(1,2)\\ \text{c.}P(4,-4),Q\text{are points on parabola}{y}^2=4x& \\ \text{such that area of}\Delta POQ\text{is 6 sq. units where}& \\ O\text{is the vertex, then coordinates of}Q\text{may be}& \text{r.}(-2,1)\\ \text{d. The chord of contact w.r.t any point on the}& \\ \text{directrix of the parabola}(y-2{)}^2=4x\text{passes}& \\ \text{through the point}& \text{s.}(4,4)\end{array}$

Which of the following is correct option

1. $a\to q,r,b\to q,c\to p,q,d\to s$
2. $a\to q,s,b\to r,c\to p,qd\to q$
3. $a\to q,b\to r,c\to p,d\to q$
4. $a\to q,s,b\to q,c\to p,s,d\to s$

Q. Let $AB$ is a normal chord of parabola $y^2=4x$, with foot of normal as $A(1,-2)$. If $AB=p\sqrt{q}$, where $p$ is a natural number and $q$ is a prime number, then $\frac{p}{q}$ is equal to