Parametric Equation of Tangent

Q. The tangents at the points $(a{t}_1^2,2a{t}_1)$, $\left(a{t}_2^2\right),2a{t}_2)$ on the parabola $y^2=4ax$ are at right angles if

1. $t_1{t}_2=-1$
2. $t_1{t}_2=1$
3. $t_1{t}_2=2$
4. $t_1{t}_2=-2$

Q.

Let $a,r,s$ and $t$ be non-zero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ and $S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is point $(2a,0)$.
If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

1. $\frac{(t^2+1{)}^2}{2t^3}$

2. $\frac{a(t^2+1{)}^2}{2t^3}$

3. $\frac{a(t^2+1{)}^2}{t^3}$

4. $\frac{a(t^2+2{)}^2}{t^3}$

Q.

The point of intersection of the tengents to the parabola $y^2=4x$ at the points, where the parameter 't' has the value 1 and 2, is

1. (3, 8)

2. (1, 5)

3. (2, 3)

4. (4, 6)

Q. Let tangent drawn to the parabola $C_1:y^2=4ax$ at $P$ meets the $y$-axis at $Q$. Another parabola $C_2$ with vertex $Q$ and focus $(0,0)$ is drawn which passes through $(2a,0).$ Identify which of the following statements can be CORRECT?

1. The equation of the curve $C_2$ is $x^2=4a(y+a)$
2. The equation of the curve $C_2$ is $x^2=-4a(y+a)$
3. Point $P$ is the extremity of latus rectum of curve $C_1$
4. Possible equation of directrix of curve $C_2$ is $y-2a=0$

Q. The portion of the tangent intercepted between the point of contact and the directrix of the parabola \( y^2 = 4ax\) subtends at the focus an angle of

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. Tangent drawn at any point on $y^2=4ax$ meets the axis of parabola at $T$ and tangent at vertex at $S$. If $TASG$ is a rectangle, where $A$ is the vertex, then locus of $G$ is

1. $y^2=ax$
2. $y^2=-ax$
3. $y^2=2ax$
4. $y^2=-2ax$

Q. If $y_1,y_2$ are the ordinates of two points P and Q on the parabola and $y_3$ is the ordinate of the point of intersection of tangents at P and Q, then

1. $y_1,y_2,y_3$ are in A.P
2. $y_1,y_3,y_2$ are in A.P
3. $y_1,y_2,y_3$ are in G.P
4. $y_1,y_3,y_2$ are in G.P