Let $$P(a, b)$$ be a point on the parabola $$y^{2}=8 x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^{2}+y^{2}-10 x-14 y+65=0$$. Let $$A$$ be the product of all possible values of $$a$$ and $$B$$ be the product of all possible values of $$b$$. Then the value of $$A+B$$ is equal to :
JEE · Math · previous-year question
- A.0
- B.25
- C.40
- D.65correct
Answer
D. 65
Explanation
Centre of circle $${x^2} + {y^2} - 10x - 14y + 65 = 0$$ is at (5, 7). Let the equation of tangent to $${y^2} = 8x$$ is $$yt = x + 2{t^2}$$ which passes through (5, 7) $$7t = 5 + 2{t^2}$$ $$ \Rightarrow 2{t^2} - 7t + 5 = 0$$ $$t = 1,{5 \over 2}$$ $$A = 2 \times {1^2} \times 2 \times {\left( {{5 \over 2}} \right)^2} = 25$$ $$B = 2 \times 2 \times 1 \times 2 \times 2 \times {5 \over 2} = 40$$ $$A + B = 65$$
Practice more JEE questions
Answer thousands more real JEE previous-year questions free, get graded instantly, and climb the global ranked leaderboard.
Practice JEE free →More JEE Math questions
- What is the total number of distinct x \in \mathbb{R} for which \left|\begin{array}{ccc}x …
- Let m be the smallest positive integer such that the coefficient of x^{2} in the expansion…
- What is the total number of distinct x \in[0,1] for which \int_{0}^{x} \frac{t^{2}}{1+t^{4…
- Let \alpha, \beta \in \mathbb{R} be such that \lim _{x \rightarrow 0} \frac{x^{2} \sin (\b…
- Let z=\frac{-1+\sqrt{3} i}{2}, where i=\sqrt{-1}, and r, s \in\{1,2,3\}. Let P=\left[\begi…
- For how many values of p, the circle x^{2}+y^{2}+2 x+4 y-p=0 and the coordinate axes have …
- Let f: \mathbb{R} \rightarrow \mathbb{R} be a differentiable function such that f(0)=0, f\…
- For a real number \alpha, if the system \[ \left[\begin{array}{ccc} 1 & \alpha & \alpha^{2…