Equation of Tangent When a Point Is Given

Q. A circle touches the parabola $y^2=4x$ at $M(1,2)$ and also touches its directrix. The $y$-coordinate of the point of contact of the circle and the directrix is

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

Q.

The point on the curve$y^2=x$, the tangent at which makes an angle $45°$ with $x-$axis is

1. $\left(\frac{1}{4},\frac{1}{2}\right)$

2. $\left(\frac{1}{2},\frac{1}{4}\right)$

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

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

Q.

If the line $m(x-1)-m^2(y-1)+1=0$ is a tangent to a parabola for any real value of m, then the equation of the parabola is

1. $x^2+2x+4y+3=0$
2. $(y-1{)}^2=4(x-1)$
3. $x^2-2x+4y-3=0$
4. $(y-1{)}^2=-4(x-1)$

Q. An equation of a tangent drawn to the curve $y=x^2-3x+2$ from the point $(1,-1)$ is

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

Q.

Mutually perpendicular tangents $\mathrm{TA}$ and $\mathrm{TB}$ are drawn to $y^2=4ax$, minimum length of $\mathrm{AB}$ is equal to ____.

Q.

The vertex and focus of a parabola are (1, 2) and (1, -1). Then the equation of the tangent at the vertex is

1. x+2y-6=0

2. x+2y-9=0

3. x+2y-4=0

4. x+2y-7=0

Q. Let $A$ be the point where the curve $5{\alpha }^2{x}^3+10\alpha x^2+x+2y-4=0(\alpha \in R,\alpha \ne 0)$ meets the $y-$axis, then the equation of the tangent to the curve at the point where normal at $A$ meets the curve again is :

1. $x=1$
2. $y=2x+2$
3. $x+2y-4=0$
4. None of these

Q. If at x = 1, y = 2x is tangent to the parabola $y=a{x}^2+bx+c$, then respective values of a, b, c possible are

1. 
2. 
3. 
4. 

Q. The equation of tangent to the parabola $y^2=x$ at point $(4,2)$ is

1. $4y+x+12=0$
2. $4y-x-4=0$
3. $2y-3x+8=0$
4. $2y+x-8=0$

Q. If the normals at three points $P,Q,R$ of the parabola $y^2=4ax$ meet at $(b,c)$ then the centriod of the triangle $PQR$ lies on the

1. axis of the parabola
2. directrix of the parabola
3. tangent at the vertex
4. director circle

Q. Let $P$ and $R$ are two points on the parabola $y=x^2$ such that $PS$ and $RS$ be the tangents at $P$ and $R,$ and $PQ$ and $RQ$ be the normal drawn at $P$ and $R$ respectively. If the length of rectangle $PQRS$ form is twice of its width, then which of the following is/are correct?

1. The possible coordinates of $S$ are $(\frac{1}{4},\frac{1}{16})$ or $(-\frac{1}{4},\frac{1}{16})$
2. The possible coordinates of $S$ are $(\frac{3}{8},-\frac{1}{4})$ or $(-\frac{3}{8},-\frac{1}{4})$
3. Area of the rectangle $PQRS$ is $\frac{125}{128}$
4. Length of the rectangle is $\frac{5\sqrt{5}}{8}$

Q. If tangent to the parabola $y^2=4x$ is also normal to $x^2=4by$ and $|b|\le \frac{1}{\sqrt{k}}$, then the numerical value of $k$ is

Q.

The point on the curve $x^2+y^2=a^2,y\ge 0$ at which the tangent is parallel to $x$ axis

1. $\left(a,0\right)$

2. $\left(-a,0\right)$

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

4. $\left(0,a\right)$

5. $\left(0,a^2\right)$

Q. If the line $y=mx+1$ is tangent to the parabola $y^2=4x$ then find the value of $m$.

Q.

What's the equation of a tangent on the parabola
$y^2=5x$ at the point (5, 5)

1. x + 2y + 5 = 0

2. x - 5y + 2 = 0

3. x - 2y + 5 = 0

4. x + 5y - 2 = 0

Q. If tangent at each point to the curve $y=2x^3+a{x}^2+6x+5$ is inclined at an acute angle, then the number of integral values of $a$ is

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Tangents are drawn from the point}(2,3)& \text{(P)}& (9,-6)\\ & \text{to the parabola}{y}^2=4x.\text{Then point(s) of}& & \\ & \text{contact is (are)}& & \\ \text{(B)}& \text{From a point}P\text{on the circle}{x}^2+y^2=5,& \text{(Q)}& (1,2)\\ & \text{the equation of chord of contact to the}& & \\ & \text{parabola}{y}^2=4x\text{is}y=2(x-2).\text{Then}& & \\ & \text{the coordinates of}P\text{are}& & \\ \text{(C)}& P(4,-4),Q\text{are points on parabola}& \text{(R)}& (-2,1)\\ & y^2=4x\text{such that area of}△POQ\text{is}6& & \\ & \text{sq. units where}O\text{is the vertex. Then}& & \\ & \text{coordinates of}Q\text{may be}& & \\ \text{(D)}& \text{The common chord of circle}{x}^2+y^2=5& \text{(S)}& (4,4)\\ & \text{and parabola}6y=5x^2+7x\text{will pass}& & \\ & \text{through point(s)}& & \\ & & \text{(T)}& (-2,2)\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(P)},\text{(S)}$
2. $\text{(A)}\to \text{(Q)},\text{(T)}$
3. $\text{(B)}\to \text{(R)}$
4. $\text{(B)}\to \text{(Q)},\text{(R)}$

Q.

The equation of normal at $(at,\frac{a}{t})$ to the hyperbola $xy=a^2$ is

1. 

2. 

3. 

4. 

Q. The triangle formed by the tangents to a parabola $y^2=4x$ at the ends of the latus rectum and the double ordinate through the focus is

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

Q. A tangent is drawn at a point on the parabola $y^2=10x$. If this tangent intersect the parabola $y^2=10x+p$ at 2 points; then,

1. p = 0

2. p > 0

3. p < 0

4. -10 < p < 10

Q. Let $(5,6)$ and $(9,-4)$ be two points on a parabola . The point of intersection of tangents at these points is $(1,-2).$ Then the slope of directrix of the parabola is

1. $1$
2. $-1$
3. $2$
4. $-2$

Q. Assertion A: Orthocentre of the triangle formed by any three tangents to the parabola lies on the directrix of the parabola.
Reason R: The orthocentre of the triangle formed by the tangents at $t_1,t_2,t_3$ to the parabola $y^2=4ax$ is $(-a,a(t_1+t_2+t_3+t_1{t}_2{t}_3\left)\right)$

1. Both A and R are true and R is not the correct explanation of A
2. Both A and R are true and R is the correct explanation of A
3. A is false but R is true
4. A is true but R is false

Q. Show that the common tangent to the parabola $y^2=4ax$ and $x^2=4by$ is $x{a}^{1/3}+y{b}^{1/3}+a^{2/3}{b}^{2/3}=0$

Q.

What's the equation of a tangent on the parabola
$y^2=5x$ at the point (5, 5)

1. x - 5y + 2 = 0

2. x - 2y + 5 = 0

3. x + 5y - 2 = 0

4. x + 2y + 5 = 0

Q. An equation of a tangent drawn to the curve $y=x^2-3x+2$ from the point $(1,-1)$ is

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

Q. If $y=2x-3$ is a tangent to the parabola $y^2=4a(x-\frac{1}{3})$, then 'a' is equal to

1. $\frac{22}{3}$
2. $-1$
3. $\frac{14}{3}$
4. $\frac{-14}{3}$

Q. Sketch the graphs of the curves $y^2=x$ and $y^2=4-3x$ and find the area enclosed between them

Q. If the tangents of a parabola at the points (x', y') and (x", y") meet at the point $(x_1,y_1)$ and the normals at the same points in $(x_2,y_2)$ , Prove that :
$x_1=\frac{y^{\prime }{y}^{\prime \prime }}{4a}$ and $y_1=\frac{y^{\prime }+y"}{2},$

Q. A circle passes through origin and meets the axes at $A$ and $B$ so that $(2,3)$ lies on $\overline{AB}$ then the of centerid of $△OAB$ is

1. $2x-3y=6xy$
2. $2x+3y=6xy$
3. $3x-2y=3xy$
4. $3x+2y=3xy$

Q.

P(t = 2) is a point on the parabola $y^2=4ax$. What is the point of the intersection of tangent at P and the directrix of the parabola?

1. (15, 10)

2. (0, 0)

3. (10, 10)

4. (-10, 15)