If tangents are drawn to the ellipse , then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is [2004]

Question

If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ , then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is [2004]

Smart Achievers · Accepted Answer

(1)

$Any tangent to ellipse \frac{x^{2}}{2} + \frac{y^{2}}{1} = 1 is$

$\frac{x \cos θ}{\sqrt{2}} + y \sin θ = 1$ $which makes intercept A B between the coordinate axes.$

$∴$ $A (\sqrt{2} \sec θ, 0), B (0, \cos e c θ)$

$If (h, k) be the mid-point of A B, then$

$2 h = \sqrt{2} \sec θ, 2 k = c o s e c θ$

$\Rightarrow \cos θ = \frac{1}{\sqrt{2} h}, \sin θ = \frac{1}{2 k}$

$Now \cos^{2} θ + \sin^{2} θ = 1$

$\Rightarrow {(\frac{1}{\sqrt{2} h})}^{2} + {(\frac{1}{2 k})}^{2} = 1$ $\Rightarrow \frac{1}{2 h^{2}} + \frac{1}{4 k^{2}} = 1$

$∴$ $Required locus is \frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$

Solution

Q.
If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ , then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is [2004]

1 $\frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$

2 $\frac{1}{4 x^{2}} + \frac{1}{2 y^{2}} = 1$

3 $\frac{x^{2}}{2} + \frac{y^{2}}{4} = 1$

4 $\frac{x^{2}}{4} + \frac{y^{2}}{2} = 1$

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Solution

Q. If tangents are drawn to the ellipse x2+2y2=2, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is [2004]

1 12x2+14y2=1

2 14x2+12y2=1

3 x22+y24=1