Let ABCD be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides AD and BC of the trapezium be parallel to the y-axis. If the diagonal AC is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of ABCD is:
Show Hint
- \( \frac{125}{8} \)
- \( \frac{75}{8} \)
- \( \frac{25}{2} \)
- \( \frac{75}{4} \)
The Correct Option is B
Solution and Explanation
We are given the geometry of the trapezium and need to calculate its area.
Step 1: First, determine the coordinates of the vertices of the trapezium using the equation \( y^2 = 4x \).
Step 2: Calculate the length of diagonal AC by using the distance formula between the points.
Step 3: Use the area formula for a trapezium, which involves calculating the parallel sides' lengths and height, to find the area.
Final Conclusion: The area of ABCD is \( \frac{75}{8} \), which is Option 2.
Top Questions on Coordinate Geometry
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
- JEE Advanced - 2025
- Mathematics
- Coordinate Geometry
- If \(a = 1 + \sum_{r=1}^{6} (-3)^{r-1} \binom{12}{2r-1}\), then the distance of the point \((12, \sqrt{3})\) from the line \(\alpha x - \sqrt{3}y + 1 = 0\) is:
- JEE Main - 2025
- Mathematics
- Coordinate Geometry
- The equation of the circle passing through the point \( (1, 2) \) and touching both axes is:
- NIMCET - 2025
- Mathematics
- Coordinate Geometry
- Let $ S $ denote the locus of the midpoints of those chords of the parabola $ y^2 = x $, such that the area of the region enclosed between the parabola and the chord is $ \frac{4}{3} $. Let $ \mathcal{R} $ denote the region lying in the first quadrant, enclosed by the parabola $ y^2 = x $, the curve $ S $, and the lines $ x = 1 $ and $ x = 4 $.
Then which of the following statements is (are) TRUE?- JEE Advanced - 2025
- Mathematics
- Coordinate Geometry
Let $ S $ denote the locus of the point of intersection of the pair of lines $$ 4x - 3y = 12\alpha,\quad 4\alpha x + 3\alpha y = 12, $$ where $ \alpha $ varies over the set of non-zero real numbers. Let $ T $ be the tangent to $ S $ passing through the points $ (p, 0) $ and $ (0, q) $, $ q > 0 $, and parallel to the line $ 4x - \frac{3}{\sqrt{2}} y = 0 $.
Then the value of $ pq $ is- JEE Advanced - 2025
- Mathematics
- Coordinate Geometry
Questions Asked in JEE Main exam
- Three infinitely long wires with linear charge density \( \lambda \) are placed along the x-axis, y-axis and z-axis respectively. Which of the following denotes an equipotential surface?
- JEE Main - 2025
- electrostatic potential and capacitance
- The number of 6-letter words, with or without meaning, that can be formed using the letters of the word "MATHS" such that any letter that appears in the word must appear at least twice is:
- JEE Main - 2025
- Calculus
- Number of functions \( f: \{1, 2, \dots, 100\} \to \{0, 1\} \), that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
- JEE Main - 2025
- Vectors
- Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
- JEE Main - 2025
- Sequences and Series
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression