Q1 The Parabola Y² = 4ax Has A Chord Perpendicular To Its Axis Whose Midpoint Is (24) Find Its Equation

Q1. The parabola y² = 4ax has a chord perpendicular to its axis whose midpoint is (2,4). Find its equation. Q2. A tangent drawn to the parabola y² = 12x meets the x-axis at A. If the tangent makes an angle of 45° with the axis, find the coordinates of A. Q3. For the ellipse x²/25 + y²/9 = 1, a normal at point P meets the major axis at N. Find the length PN. Q4. A line intersects the hyperbola x²/16 − y²/9 = 1 at points P and Q. Find the coordinates of P and Q if the line passes through (5,2). Q5. Find the equation of the tangent to the parabola y² = 20x which passes through point (8,4). Q6. The ellipse x²/16 + y²/9 = 1 has a tangent which intersects the axes at A and B. Find the locus of the midpoint of AB. Q7. A hyperbola x²/9 − y²/4 = 1 has a chord perpendicular to the transverse axis passing through (3,2). Find its equation. Q8. Find the points on the parabola y² = 16x at which the tangents are inclined at 60° to the axis. Q9. Find the eccentricity of the ellipse whose foci are at (±4,0) and whose minor axis length is 6. Q10. From point (6,3), tangents are drawn to the hyperbola x²/16 − y²/9 = 1. Find the angle between the tangents. Q11. Find the equation of the chord of the parabola y² = 12x which passes through (3,6) and has slope −1. Q12. The ellipse x²/36 + y²/16 = 1 has a chord PQ which subtends 90° at the center. Find the length of PQ. Q13. Find the locus of the intersection points of perpendicular tangents drawn to the ellipse x²/9 + y²/4 = 1. Q14. A tangent to the parabola x² = 16y intersects the axes at points A and B. Find the locus of the midpoint of AB. Q15. The hyperbola x²/25 − y²/16 = 1 has a chord joining points P(5,3) and Q(−5,−3). Find its equation. Q16. (Application) A parabolic dish y² = 20x is used to reflect radio waves. Determine whether a signal from point (10,10) reaches the focus. Q17. (Application) A hall has an elliptical floor plan x²/25 + y²/9 = 1. A person at one focus speaks. Find the distance the sound travels to reach the other focus. Q18. (Application) A parabolic arch bridge has a span of 30 m and maximum height of 12 m. Vertex at origin. Find the equation of the parabola and the tangent at the base. Q19. (Application) A satellite moves along x² = 24y. Find its distance from the focus when at (6,9). Q20. (Application) A searchlight is at the focus of the ellipse x²/16 + y²/4 = 1. Determine whether it can illuminate the point (2,1). Q21. (Application) A cooling tower has cross-section modeled by x²/9 − y²/4 = 1. Check whether point (5,2) is inside, on, or outside. Q22. (Application) A parabolic car headlight y² = 20x has the bulb at the focus. Find the distance of the bulb from the vertex. Q23. (Application) A sound reflector parabola has focus 2 m from vertex. Verify whether points (4,4) and (9,6) lie on it. Q24. (Application) A billboard frame shaped like y²/25 − x²/16 = 1. Find the shortest distance from the origin to a point on the hyperbola. Q25. (Application) A planet moves along an ellipse with foci at (6,0) and (−6,0). If its position is P(5,8), determine whether it lies on an ellipse with major axis length 20.

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