If the locus of the point, whose distances from the point $(2, 1)$ and $(1, 3)$ are in the ratio $5 : 4$, is \[ ax^2 + by^2 + cxy + dx + ey + 170 = 0, \] then the value of $a^2 + 2b + 3c + 4d + e$ is equal to:
- 5
- -27
- 37
- 437
The Correct Option is C
Solution and Explanation
Let \( P(x, y) \)
\[ \frac{(x - 2)^2 + (y - 1)^2}{(x - 1)^2 + (y - 3)^2} = \frac{25}{16} \]
Expanding and simplifying:
\[ 9x^2 + 9y^2 + 14x - 118y + 170 = 0 \]
From the equation:
\[ a^2 + 2b + 3c + 4d + e = 81 + 18 + 0 + 56 - 118 \]
Calculating:
\[ = 155 - 118 \]
\[ = 37 \]
Top Questions on 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 $ \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
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
- 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
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Complex numbers
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
- 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