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Hyperbola question

2021 · 25 Feb · Shift 1 · Q37
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  5. /2021 · 25 Feb · Shift 1 · Q37

Hyperbola question

2021 · 25 Feb · Shift 1 · Q37

JEE MainMathematicsHyperbolaNumerical+4 / −1
The locus of the point of intersection of the lines (3)kx+ky−43=0\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0(3​)kx+ky−43​=0 and 3x−y−4(3)k=0\sqrt 3 x - y - 4\left( {\sqrt 3 } \right)k = 03​x−y−4(3​)k=0 is a conic, whose eccentricity is ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 2

  1. Write the given lines

The two lines are

3 kx+ky−43=0\sqrt{3}\,k x + ky - 4\sqrt{3}=03​kx+ky−43​=0

and

3x−y−43k=0.\sqrt{3}x - y - 4\sqrt{3}k=0.3​x−y−43​k=0.

We need the locus of their point of intersection as the parameter kkk varies.

  1. Find the intersection point (x,y)(x,y)(x,y) in terms of kkk

From the second line,

3x−y=43k.\sqrt{3}x - y = 4\sqrt{3}k.3​x−y=43​k.

So,

k=3x−y43.k = \frac{\sqrt{3}x-y}{4\sqrt{3}}.k=43​3​x−y​.

Now substitute this into the first line:

k(3x+y)=43.k(\sqrt{3}x+y)=4\sqrt{3}.k(3​x+y)=43​.

Using the value of kkk,

3x−y43(3x+y)=43.\frac{\sqrt{3}x-y}{4\sqrt{3}}(\sqrt{3}x+y)=4\sqrt{3}.43​3​x−y​(3​x+y)=43​.

Multiply both sides by 434\sqrt{3}43​:

(3x−y)(3x+y)=16⋅3=48.(\sqrt{3}x-y)(\sqrt{3}x+y)=16\cdot 3=48.(3​x−y)(3​x+y)=16⋅3=48.

Now use difference of squares:

(3x)2−y2=48(\sqrt{3}x)^2-y^2=48(3​x)2−y2=48 3x2−y2=48.3x^2-y^2=48.3x2−y2=48.

Thus the locus is

3x2−y2=48.3x^2-y^2=48.3x2−y2=48.
  1. Identify the conic

Rewrite:

x216−y248=1.\frac{x^2}{16}-\frac{y^2}{48}=1.16x2​−48y2​=1.

This is a hyperbola of the form

x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1a2x2​−b2y2​=1

with

a2=16,b2=48.a^2=16,\qquad b^2=48.a2=16,b2=48.
  1. Find the eccentricity

For the hyperbola

x2a2−y2b2=1,\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,a2x2​−b2y2​=1,

the eccentricity is

e=1+b2a2.e=\sqrt{1+\frac{b^2}{a^2}}.e=1+a2b2​​.

So,

e=1+4816=1+3=4=2.e=\sqrt{1+\frac{48}{16}}=\sqrt{1+3}=\sqrt{4}=2.e=1+1648​​=1+3​=4​=2.
  1. Final answer

The eccentricity of the conic is

2.\boxed{2}.2​.
  1. Comparison with stored correct answer

Stored correct answer: 222

My derived answer is also 222, so they agree.

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