Shortest Distance between Two Curves

Q. The shortest distance between the curves $y^2=x^3$ and $9x^2+9y^2–30y+16=0$ is

1. $\frac{\sqrt{13}}{3}$
2. $\frac{2}{3}$
3. $\frac{\sqrt{13}}{3}-1$
4. $\frac{5}{3}$

Q. The shortest distance between line $y-x=1$ and curve $x=y^2$ is

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

Q. If $P$ is a point on the parabola $y=x^2+4$ which is closest to the straight line $y=4x-1,$ then the co-ordinates of $P$ are

1. $(-2,8)$
2. $(1,5)$
3. $(3,13)$
4. $(2,8)$

Q.

Maximum slope of the curve $y=-x^3+3x^2+9x-27$ is

1. $0$

2. $12$

3. $16$

4. $32$

Q. The shortest distance between the line $x-y=1$ and the curve $x^2=2y$ is :

1. $\frac{1}{2}$
2. $\frac{1}{2\sqrt{2}}$
3. $\frac{1}{\sqrt{2}}$
4. $0$

Q. A helicopter is flying along the curve given by $y-x^{\frac{3}{2}}=7,(x\ge 0)$. A soldier positioned at the point $(\frac{1}{2},7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

1. $\frac{\sqrt{5}}{6}$
2. $\frac{1}{2}$
3. $\frac{1}{3}\sqrt{\frac{7}{3}}$
4. $\frac{1}{6}\sqrt{\frac{7}{3}}$

Q. The line x + y + 2 =0 is a tangent to a parabola $y^2=4ax$ at the point A. If it intersect the directrix at B and tangent at vertex at C, then

1. $AB=\sqrt{32}$
2. $AC=4$
3. $AC.BC=8$
4. Area of $\Delta ABC$ = 5 sq. units.

Q. From a variable point $P,$ normal is drawn to the hyperbola $xy=16.$ If sum of slopes of normal is equal to the sum of ordinates and abscissas of feet of normal, then the locus of point $P$ is the curve $C.$ Which of the following is (are) CORRECT ?

1. $C$ is a parabola of length of latus rectum $16$ units
2. $C$ is an ellipse of length of latus rectum $16$ units
3. $C$ is a hyperbola of length of latus rectum $16$ units
4. Focus of $C$ is at $(8,0)$

Q.

India is planning to launch a rocket with a destroyer to protect earth from a comet approaching earth. Scientists want to fire the destroyer when the rocket and the comet are closest in order to have maximum impact. For this, they found the equation of the path the rocket ($y+x-2=0$ ) will follow and the path of the comet near earth
( $y=x^2+3x+2$) with respect to a reference point [(0, 0)]. Find the distance between the reference point and the point where the destroyer would hit the comet.
(Assume that the rocket can be controlled to reach closest point at the same time the comet reaches )

1. 1
2. -1
3. $\sqrt{2}$
4. 0

Q. A helicopter is flying along the curve given by $y-x^{\frac{3}{2}}=7,(x\ge 0)$. A soldier positioned at the point $(\frac{1}{2},7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

1. $\frac{\sqrt{5}}{6}$
2. $\frac{1}{2}$
3. $\frac{1}{3}\sqrt{\frac{7}{3}}$
4. $\frac{1}{6}\sqrt{\frac{7}{3}}$

Q. The shortest distance between the curves $y^2=x^3$ and $9x^2+9y^2-30y+16=0$ is

1. $\frac{\sqrt{13}}{3}$
2. $\frac{2}{3}$
3. $\frac{5}{3}$
4. $\frac{\sqrt{13}}{3}-1$

Q. Find the slope of the tangent to curve $y=x^3-x+1$ at the point whose $x$-coordinate is $2$.

Q.

C1 and C2 are two non-intersecting curves. Common normal of C1 and C2 intersect at the points (1, 0) and (4, 4) on the curves. If the slope of the normal is 30 degrees, find the shortest distance between the curves assuming they have only one common normal

1. $5\sin \left(30°\right)$

2. $5\cos \left(30°\right)$

3. 5

4. 
   $5\tan \left(30°\right)$

Q. The shortest distance between line $y-x=1$ and curve $x=y^2$ is

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

Q. The shortest distance between the curves $y^2=x^3$ and $9x^2+9y^2-30y+16=0$ is

1. $\frac{\sqrt{13}}{3}$
2. $\frac{2}{3}$
3. $\frac{5}{3}$
4. $\frac{\sqrt{13}}{3}-1$

Q.

India is planning to launch a rocket with a destroyer to protect earth from a comet approaching earth. Scientists want to fire the destroyer when the rocket and the comet are closest in order to have maximum impact. For this, they found the equation of the path the rocket ($y+x-2=0$ ) will follow and the path of the comet near earth
( $y=x^2+3x+2$) with respect to a reference point [(0, 0)]. Find the distance between the reference point and the point where the destroyer would hit the comet.
(Assume that the rocket can be controlled to reach closest point at the same time the comet reaches )

1. 1
2. -1
3. 0
4. $\sqrt{2}$

Q. If $P$ is a point on the parabola $y=x^2+4$ which is closest to the straight line $y=4x-1,$ then the co-ordinates of $P$ are

1. $(-2,8)$
2. $(1,5)$
3. $(3,13)$
4. $(2,8)$

Q. If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

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

Q. A horse runs along a circle with a speed of $20km/h$. A lantern is at the centre of the circle. A fence is there along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse moves along the fence at the moment when it covers $\frac{1}{8}$ of the circle in $km/h$ is

1. $40$
2. $30$
3. $60$
4. $20$

Q. The length of the sub-tangent, ordinate and the sub-normal are in

1. A.P.
2. H.P.
3. G.P.
4. Arithmetico geometric progression

Q. Find the point on the curve $x^2=2y$ which is closest to the point $(0,5).$

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

Q. The area bounded by two branches of the curve $(y-x{)}^2=x^3\&x=1$ equals

1. $6/5$
2. $4/5$
3. $3/5$
4. $5/4$

Q. $y=mx+1$ be a tangent to $y^2=4x$ then find the value of $m$.

Q. The shortest distance between line $y-x=1$ and curve $x=y^2$ is

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

Q.

C1 and C2 are two non-intersecting curves. Common normal of C1 and C2 intersect at the points (1, 0) and (4, 4) on the curves. If the slope of the normal is 30 degrees, find the shortest distance between the curves assuming they have only one common normal

1. 5

2. 
   $5\tan \left(30°\right)$

3. $5\sin \left(30°\right)$

4. $5\cos \left(30°\right)$

Q.

C1 and C2 are two non-intersecting curves. Common normal of C1 and C2 intersect at the points (1, 0) and (4, 4) on the curves. If the slope of the normal is 30 degrees, find the shortest distance between the curves assuming they have only one common normal

1. 5

2. 
   $5\tan \left(30°\right)$

3. $5\sin \left(30°\right)$

4. $5\cos \left(30°\right)$

Q.

India is planning to launch a rocket with a destroyer to protect earth from a comet approaching earth. Scientists want to fire the destroyer when the rocket and the comet are closest in order to have maximum impact. For this, they found the equation of the path the rocket ($y+x-2=0$ ) will follow and the path of the comet near earth
( $y=x^2+3x+2$) with respect to a reference point [(0, 0)]. Find the distance between the reference point and the point where the destroyer would hit the comet.
(Assume that the rocket can be controlled to reach closest point at the same time the comet reaches )

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

Q.

India is planning to launch a rocket with a destroyer to protect earth from a comet approaching earth. Scientists want to fire the destroyer when the rocket and the comet are closest in order to have maximum impact. For this, they found the equation of the path the rocket ($y+x-2=0$ ) will follow and the path of the comet near earth
( $y=x^2+3x+2$) with respect to a reference point [(0, 0)]. Find the distance between the reference point and the point where the destroyer would hit the comet.
(Assume that the rocket can be controlled to reach closest point at the same time the comet reaches )

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

Q. Consider the two orbits around the sun shown below. Orbit P is circular with radius R, orbit Q is elliptical such that the farthest point b is between 2R and 3R, and the nearest point a is between R/3 and R/2 . Consider the magnitudes of the velocity of the circular orbit $v_c$ , the velocity of the comet in the elliptical orbit at the farthest point $v_b$ , and the velocity of the comet in the elliptical orbit at the nearest point $v_a$ . Which of the following ranking is correct?

1. $v_b>v_c>2v_a$
2. $2v_c>v_b>v_a$
3. $10v_b>v_a>v_c$
4. $v_c>v_a>4v_b$

Q. Find the curve $y=f\left(x\right)$ where $f\left(x\right)\ge 0,f\left(0\right)=0$ bounding a curvilinear trapezoid with the base [0, x] whose area is proportional to $(n+1{)}^{th}$ power of $f\left(x\right)$. It is known that $f\left(1\right)=1$.

1. $y=x^{-n}$
2. $y=x^{-1/n}$
3. $y=x^n$
4. $y=x^{1/n}$