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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Hyperbola

NTA Abhyas 2020 · Mathematics · Hyperbola

The locus of the mid-points of the chords of the hyperbola 3x^ 2 -...

The locus of the mid-points of the chords of the hyperbola 3x2−2y2+4x−6y=03x^{2} - 2y^{2} + 4x - 6y = 0 which are parallel to the line y=2x+4y = 2x + 4& is

Options

  1. A3x−4y=43x - 4y = 4
  2. B4x−4y=34x - 4y = 3
  3. C3y−4x+4=03y - 4x + 4 = 0
  4. D3x−4y=23x - 4y = 2

Correct answer

A. 3x−4y=43x - 4y = 4

Step-by-step solution

Let mid-point of the chord is (h,k)\left(h, k\right)
⇒⇒ equation of the chord is
3hx−2ky+4(x+h2)−6(y+k2)=3h2−2k2+4h−6k3hx - 2ky + 4\left(\dfrac{x + h}{2}\right) - 6\left(\dfrac{y + k}{2}\right) = 3h^{2} - 2k^{2} + 4h - 6k
⇒(3h+2)x+(−2k−3)y=3h2−2k2+2h−3k⇒ \left(3h + 2\right)x + \left(- 2k - 3\right)y = 3h^{2} - 2k^{2} + 2h - 3k
⇒⇒ Slope of the chord is equal to 3h+22k+3=2\dfrac{3h + 2}{2k + 3} = 2
⇒⇒ locus of the mid-points is 3x+2=4y+63x + 2 = 4y + 6
⇒3x−4y−4=0⇒ 3x - 4y - 4 = 0
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