Let the tangent drawn to the parabola y2= 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola <math><mrow><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>β</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math> at the point ( α+ 4, β+ 4) does NOT pass through the point:

Any tangent to y2 = 24x at (α, β) is βy = 12 (x + α) therefore Slope = 12βand perpendicular to 2x + 2y = 5 =>12 =β and α= 6 Hence hyperbola is x262−y2122 = 1 and normal is drawn at (10, 16)therefore equation of normal 36x10+144y16=36+144⇒x50+y20=1 This does not pass through (15, 13) out of given option.

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Maths Class 12th Vector Algebra Maths NCERT Exemplar Solutions Class 12th Chapter One

Let the tangent drawn to the parabola y²= 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola $\frac{x^{2}}{α^{2}} - \frac{y^{2}}{β^{2}} = 1$ at the point ( α+ 4, β+ 4) does NOT pass through the point:

Option 1 -

(25, 10)

Option 2 -

(20, 12)

Option 3 -

(30, 8)

Option 4 -

(15, 13)

5 Views | Posted 2 months ago

Asked by Shiksha User

1 Answer

Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 4

Detailed Solution:

Any tangent to y² = 24x at (α, β) is βy = 12 (x + α) therefore Slope = $\frac{12}{β}$

and perpendicular to 2x + 2y = 5 =>12 =β and α= 6 Hence hyperbola is $\frac{x^{2}}{6^{2}} - \frac{y^{2}}{1 2^{2}}$ = 1 and normal is drawn at (10, 16)

therefore equation of normal $\frac{36 x}{10} + \frac{144 y}{16} = 36 + 144 \Rightarrow \frac{x}{50} + \frac{y}{20} = 1$ This does not pass through (15, 13) out of given option.

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Let the tangent drawn to the parabola y²= 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola $\frac{x^{2}}{α^{2}} - \frac{y^{2}}{β^{2}} = 1$ at the point ( α+ 4, β+ 4) does NOT pass through the point:

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Let the tangent drawn to the parabola y2= 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola x2α2−y2β2=1 at the point ( α+ 4, β+ 4) does NOT pass through the point:

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Let the tangent drawn to the parabola y²= 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola $\frac{x^{2}}{α^{2}} - \frac{y^{2}}{β^{2}} = 1$ at the point ( α+ 4, β+ 4) does NOT pass through the point: