Parametric Equation of Tangent

Q.

If the tangents at P and Q in the parabola meet in T, then which of the following statements are correct.

1. TP and TQ subtends equal angle at the focus S

2. $S{T}^2$ = SP.SQ

1. Only 1

2. Both 1 2

3. None of these

4. Only 2

Q.

The angle between the tangents at those points on the curve $x=t^2+1$ and $y=t^2-t-6$, where it meets $X-$ axis is

1. $\pm {\tan }^{-1}\left(\frac{4}{29}\right)$

2. $\pm {\tan }^{-1}\left(\frac{5}{49}\right)$

3. $\pm {\tan }^{-1}\left(\frac{10}{49}\right)$

4. $\pm {\tan }^{-1}\left(\frac{8}{29}\right)$

Q.

If the tangents are drawn from (3, 2) to the hyperbola $x^2-9y^2=9$. Find the area of the triangle (in sq. unit) that these tangents form with their chord of contact.

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. 
2. 
3. 
4. 

Q. If a circle $S=0$ touches the parabola $y^2=4x$ at the point $(1,-2)$ and is passing through the origin, then the radius of $S=0$ is equal to

1. $5\sqrt{2}$
2. $\frac{5}{\sqrt{2}}$
3. $2\sqrt{5}$
4. $\frac{2}{\sqrt{5}}$

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. 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. are in A.P
2. are in A.P
3. are in G.P
4. are in G.P

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. 
2. 
3. 
4. 

Q. Two tangents are drawn to the curve $4x^2-9y^2=36$ such that the product of their slopes is $4$, then the locus of their point of intersection is

1. is an ellipse with $e^2=\frac{4}{5}$
2. is an ellipse with $e^2=\frac{3}{5}$
3. is an hyperbola with $e^2=\frac{5}{4}$
4. is an hyperbola with $e^2=5$

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. (4, 6)

2. (3, 8)

3. (1, 5)

4. (2, 3)

Q. I. Common tangent to the parabolas $y^2=32x$ and $x^2=108y$ is $2x+3y+36=0$
II: Common tangent to the parabolas $y^2=32x$ and $x^2=-108y$ is $2x-3y+36=0$.

1. only I
2. only II
3. neither I nor II
4. Both I and II

Q. If a circle $S=0$ touches the parabola $y^2=4x$ at the point $(1,-2)$ and is passing through the origin, then the radius of $S=0$ is equal to

1. $5\sqrt{2}$
2. $\frac{5}{\sqrt{2}}$
3. $2\sqrt{5}$
4. $\frac{2}{\sqrt{5}}$

Q. Equation of a common tangent to the curves $y^2=8x$ and $xy=-1$ is:

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

Q. Equation of a tangent to the parabola $y^2=12x$ which make an angle of ${45}^0$ with line $y=3x+77$ is

1. $2x-4y+3=0$
2. $x-2y+12=0$
3. $4x+2y+3=0$
4. $2x+y-12=0$

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

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

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

Q. Let $A$ be the point on the curve $y=x^2$ in the first quadrant. Let $B$ be the point of intersection of the tangent to the curve $y=x^2$ at the point $A$ and the $x-axis$. If the area of the region bounded by the curve $y=x^2$ and the line segment $OA$ is $\frac{p}{q}$ times the area of the triangle $OAB,$ where $O$ is the origin, then the least positive integral value of $p+q$ is (Where $p$ and $q$ are co-promes numbers)

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. The point of the contact of the tangent to the parabola $y^2=4ax$ which makes an angle of ${30}^o$ with $X$-axis, is

1. $(2\sqrt{3}a,3a)$
2. $(\sqrt{3}a,6a)$
3. $(3a,2\sqrt{3}a)$
4. None of these

Q. Find the equations of the common tangents to the parabolas $y=x^2$ and $y=-x^2+3x-2.$

Q. Let $P$ be the foot of the perpendicular from focus $S$ of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ on the line $bx-ay=0$ and let $C$ be the centre of hyperbola. Then the area of the rectangle whose sides are equal to that of $SP$ and $CP$ is

1. $2ab$
2. $ab$
3. $\frac{(a^2+b^2)}{2}$
4. $\frac{a}{b}$

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. The triangle formed by the tangents by the tangents to a parabola $y^2=4ax$ at the ends of the rectum is

1. equilateral
2. isosceles
3. right-angled isosceles
4. dependent on the value of a for its classification

Q. The equation of the tangent to the parabola $y^2=4x$ inclined at an angle of $\frac{\pi }{4}$ to the +ve direction of $x-axis$ is.

1. $x-y+4=0$
2. $x-y-1=0$
3. $x+y-4=0$
4. $x-y+1=0$

Q. A curve is represented by the equations $x={\sec }^{2}t$ and $y=\cot t,$ where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t=\frac{\pi }{4}$ meets the curve again at the point $Q,$ then which of the following is/are correct

1. Slope of tangent at $Q=-\frac{1}{16}$
2. Slope of tangent at $Q=\frac{1}{16}$
3. $PQ=\frac{5\sqrt{3}}{2}$
4. $PQ=\frac{3\sqrt{5}}{2}$

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.

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)