`P(x , y)` is a variable point on the parabola `y^2=4a x` and `Q(x+c ,y+c)` is another variable point, where `c` is a constant. The locus of the midpo

Monjur Alahi

`P(x , y)` is a variable point on the parabola `y^2=4a x` and `Q(x+c ,y+c)` is another variable point, where `c` is a constant. The locus of the midpoint of `P Q` is a/n parabola (b) hyperbola hyperbola (d) circle
A. parabola
B. ellipse
C. hyperbola
D. circle


0 like 0 dislike
4 views
Share with your friends

1 Answers

Salim Ahmed Kawsar

Call

Correct Answer - A
(1) Let the midpoint of PQ be `(alpha,beta)`. Then,
`alpha=x+(c)/(2)andbeta=y+(c)/(2)`
`or(beta-(c)/(2))^(2)=4a(alpha-(c)/(2))`
`or(y-(c)/(2))^(2)=4a(x-(c)/(2))`
which is the required locus.

0 like 0 dislike
1 views
Talk Doctor Online in Bissoy App
`A B C` is a variable triangle such that `A` is (1, 2), and `Ba n dC` on the line `y=x+lambda(lambda` is a variable). Then the locus of the orthocentr

Correct Answer - D As the altitude from A is fixed and the orthocenter lies on the altitude, x+y=3 is the required locus.

2 Answers 1 views
If the coordinates of a variable point `P` are `(acostheta,bsintheta),` where `theta` is a variable quantity, then find the locus of `Pdot`

Let `P -=(x,y)`. According to the question, `x=acostheta`(1) `y=bsintheta` (2) Sqaureing and adding (1) and (2), we get `(x^2)/(a^2)+(y^2)/(b^2)=cos^2theta+sin^2theta` or `(x^2)/(a^2)+(y^2)/(b^2)=1`

2 Answers 1 views
Find the locus of the middle points of the chords of the parabola `y^2=4a x` which subtend a right angle at the vertex of the parabola.

Let the chord joining points `P(at_(1)^(2),2at_(1))andQ(at_(2)^(2),2at_(2))` subtends right angle at vertex of the parabola `y^(2)=4ax` `:." "t_(1)t^(2)=-4` If midpoint of chord PQ is R (h,k), then `h=(at_(1)^(2)+at_(2)^(2))/(2),k=(2at_(1)+2at_(2))/(2)` `:." "(2h)/(a)=(t_(1)+t_(2))^(2)-2t_(1)t_(2)=(t_(1)+t_(2))^(2)+8and(k)/(a)-t_(1)+t_(2)` Eliminating...

2 Answers 1 views
Tangents are drawn to the parabola `y^2=4a x` at the point where the line `l x+m y+n=0` meets this parabola. Find the point of intersection of these t

Given line is `lx+my+n=0` Let this intersect parabola `y^(2)=4ax` at points P and Q. Now, tangents drawn to parabola at points P and Q, intersect at R(h,k). lt,brgt So, PQ...

2 Answers 1 views
If the chord of contact of tangents from a point `P` to the parabola `y^2=4a x` touches the parabola `x^2=4b y ,` then find the locus of `Pdot`

Chord of contact of parabola `y^(2)=4ax` w.r.t. point `P(x_(1),y_(1))` is `yy_(1)=2a(x+x_(1))`. Solving this line with parabola `x^(2)=4by`, we get `x^(2)=4bxx(2a)/(y_(1))(x+x_(1))` `ory_(1)x^(2)=8bx-8abx_(1)=0` Since the line touches the parabola, this equation will...

2 Answers 1 views
The locus of a point on the variable parabola `y^2=4a x ,` whose distance from the focus is always equal to `k ,` is equal to (`a` is parameter) `4x^2

Correct Answer - `4x^(2)+y^(2)-4kx=0` Let the point be `(at^(2),2at)` and focus S be (a,0). Now, `SP=at^(2)+a=k` (given) (1) Let `(alpha,beta)` be the moving point. Then `alpha=at^(2)andbeta=2at` `rArr" "(alpha)/(beta)=(t)/(2)anda=(beta^(2))/(4alpha)" "(because "Point" (alpha,beta)...

2 Answers 2 views
Show that the locus of a point that divides a chord of slope 2 of the parabola `y^2=4x` internally in the ratio `1:2` is parabola. Find the vertex of

Correct Answer - Vertex `-=((2)/(9),(8)/(9))` Slope of the chord joining point `P(t_(1))andQ(t_(2))` on the parabola `y^(2)=4x` is `(2)/(t_(2)+t_(1))=2` (given) `:." "t_(2)+t_(1)=1` (1) Let point R(h,k) divide PQ in the ratio 1...

2 Answers 1 views
If P be a point on the parabola `y^2=3(2x-3)` and M is the foot of perpendicular drawn from the point P on the directrix of the parabola, then length

Correct Answer - C (3) `y^(2)=6(x-(3)/(2))` The equation of directrix is `x-(3)/(2)=-(3)/(2)orx=0` Vertex is `((3)/(2),0)` and focus of P be `((3)/(2)+(3)/(2)t^(2),3t)`. Then the coordinates of M are (0,3t). Therefore, `MS=sqrt(9+9t^(2))` `MP=(3)/(2)+(3)/(2)t^(2)`...

2 Answers 1 views
Double ordinate `A B` of the parabola `y^2=4a x` subtends an angle `pi/2` at the focus of the parabola. Then the tangents drawn to the parabola at `Aa

Correct Answer - A (1) Let `A-=(at^(2),2at),B-=(at^(2),-2at)`. Then `m_(OA)=(2)/(t),m_(OB)=(-2)/(t)` Thus, `((2)/(t))((-2)/(t))=-1` `or" "t^(2)=4` Thus, the tangents will intersect at (-4a,0).

2 Answers 2 views
The locus of the midpoint of the focal distance of a variable point moving on theparabola `y^2=4a x` is a parabola whose latus rectum is half the latu

Correct Answer - A::B::C::D 1,2,3,4 Any point on the parabola is `P(at^(2),2at)`. Therefore, the midpoint of S(a,0) and `P(at^(2),2at)` is `R((a+at^(2))/(2),at)-=(h,k)` `:.h=(a+at^(2))/(2),k=at` Eliminate t, i.e., `2x=a(1+(y^(2))/(a^(2)))=a+(y^(2))/(a)` `i.e.," " 2ax=a^(2)+y^(2)` `i.e," "y^(2)=2a(x-(a)/(2))`...

2 Answers 1 views
X