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Parabola question

2022 · 28 Jun · Shift 2 · Q36
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  5. /2022 · 28 Jun · Shift 2 · Q36

Parabola question

2022 · 28 Jun · Shift 2 · Q36

JEE MainMathematicsParabolaMCQ+4 / −1
If vertex of a parabola is (2, −-− 1) and the equation of its directrix is 4x −-− 3y = 21, then the length of its latus rectum is :
  1. A
    2
  2. B
    8
  3. C
    12
  4. D
    16
View written solutionFree

Correct answer: B

  1. Use the definition of a parabola

For a parabola, the vertex lies midway between the focus and the directrix along the axis. Hence, the distance from the vertex to the directrix equals the focal length aaa.

The length of the latus rectum is 4a4a4a.

  1. Find the perpendicular distance from the vertex to the directrix

Vertex: (2,−1)(2,-1)(2,−1)

Directrix: 4x−3y=214x-3y=214x−3y=21 which can be written as 4x−3y−21=04x-3y-21=04x−3y−21=0

Distance of point (x1,y1)(x_1,y_1)(x1​,y1​) from line Ax+By+C=0Ax+By+C=0Ax+By+C=0 is d=∣Ax1+By1+C∣A2+B2d=\frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}d=A2+B2​∣Ax1​+By1​+C∣​

Here, A=4,B=−3,C=−21,(x1,y1)=(2,−1)A=4,\quad B=-3,\quad C=-21,\quad (x_1,y_1)=(2,-1)A=4,B=−3,C=−21,(x1​,y1​)=(2,−1)

So, d=∣4(2)+(−3)(−1)−21∣42+(−3)2d=\frac{|4(2)+(-3)(-1)-21|}{\sqrt{4^2+(-3)^2}}d=42+(−3)2​∣4(2)+(−3)(−1)−21∣​ =∣8+3−21∣16+9=\frac{|8+3-21|}{\sqrt{16+9}}=16+9​∣8+3−21∣​ =∣−10∣5=\frac{|-10|}{5}=5∣−10∣​ =2=2=2

Thus, the focal length is a=2a=2a=2

  1. Find the length of latus rectum

Length of latus rectum=4a=4×2=8\text{Length of latus rectum}=4a=4\times 2=8Length of latus rectum=4a=4×2=8

  1. Check options
  • A: 222 ❌
  • B: 888 ✅
  • C: 121212 ❌
  • D: 161616 ❌

Therefore, the correct answer is B.

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