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
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)
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
