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
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
- (25, 10)
- (20, 12)
- (30, 8)
- (15, 13)
The Correct Option is D
Solution and Explanation
Any tangent to y2 = 24x at (α, β)
βy = 12(x + α)
Slope=\(\frac{12}{β}\) and perpendicular to 2x+2y=5
\(\frac{12}{β}=1\)
β=12,
α=6
Hence, hyperbola is
\(\frac{x^2}{36}−\frac{y^2}{144}=1\)
and normal is drawn at (10, 16)
Equation of normal
\(\frac{36⋅x}{10}+\frac{144⋅y}{16}=\)36+144
\(=\frac{x}{50}+\frac{y}{20}=1\)
This does not pass though (15, 13) out of given option.
So, the correct option is (D): (15, 13)
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Concepts Used:
Conic Sections
When a plane intersects a cone in multiple sections, several types of curves are obtained. These curves can be a circle, an ellipse, a parabola, and a hyperbola. When a plane cuts the cone other than the vertex then the following situations may occur:
Let ‘β’ is the angle made by the plane with the vertical axis of the cone
- When β = 90°, we say the section is a circle
- When α < β < 90°, then the section is an ellipse
- When α = β; then the section is said to as a parabola
- When 0 ≤ β < α; then the section is said to as a hyperbola
Read More: Conic Sections